# Exact Priors, Likelihoods, Partition Normalization and Zero Denominators

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261102-r06-bayes-theorem-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/4, 1/2, 1/4, 1/8, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/4, 1/2, 1/4, 1/8, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0001/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 2. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/9, 1/2, 2/9, 1/9, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/9, 1/2, 2/9, 1/9, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0002/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 3. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/5, 1/2, 1/5, 1/10, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/5, 1/2, 1/5, 1/10, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0003/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 4. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/11, 1/2, 2/11, 1/11, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/11, 1/2, 2/11, 1/11, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0004/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 5. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/6, 1/2, 1/6, 1/12, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/6, 1/2, 1/6, 1/12, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0005/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 6. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/13, 1/2, 2/13, 1/13, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/13, 1/2, 2/13, 1/13, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0006/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 7. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/7, 1/2, 1/7, 1/14, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/7, 1/2, 1/7, 1/14, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0007/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 8. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/15, 1/2, 2/15, 1/15, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/15, 1/2, 2/15, 1/15, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0008/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 9. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/8, 1/2, 1/8, 1/16, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/8, 1/2, 1/8, 1/16, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0009/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 10. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/17, 1/2, 2/17, 1/17, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/17, 1/2, 2/17, 1/17, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0010/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 11. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/9, 1/2, 1/9, 1/18, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/9, 1/2, 1/9, 1/18, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0011/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 12. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/19, 1/2, 2/19, 1/19, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/19, 1/2, 2/19, 1/19, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0012/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 13. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/10, 1/2, 1/10, 1/20, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/10, 1/2, 1/10, 1/20, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0013/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 14. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/21, 1/2, 2/21, 1/21, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/21, 1/2, 2/21, 1/21, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0014/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 15. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/11, 1/2, 1/11, 1/22, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/11, 1/2, 1/11, 1/22, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0015/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 16. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/23, 1/2, 2/23, 1/23, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/23, 1/2, 2/23, 1/23, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0016/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 17. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/12, 1/2, 1/12, 1/24, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/12, 1/2, 1/12, 1/24, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0017/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 18. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/25, 1/2, 2/25, 1/25, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/25, 1/2, 2/25, 1/25, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0018/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 19. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/13, 1/2, 1/13, 1/26, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/13, 1/2, 1/13, 1/26, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0019/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 20. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/27, 1/2, 2/27, 1/27, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/27, 1/2, 2/27, 1/27, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0020/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 21. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/14, 1/2, 1/14, 1/28, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/14, 1/2, 1/14, 1/28, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0021/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 22. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/29, 1/2, 2/29, 1/29, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/29, 1/2, 2/29, 1/29, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0022/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 23. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/15, 1/2, 1/15, 1/30, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/15, 1/2, 1/15, 1/30, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0023/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 24. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(2/31, 1/2, 2/31, 1/31, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/31, 1/2, 2/31, 1/31, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0024/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 25. Prior and Likelihood Must Both Enter a Bayes Reversal

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).

View answer and explanation(1/16, 1/2, 1/16, 1/32, 1/2, not_causal)

Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/16, 1/2, 1/16, 1/32, 1/2, not_causal).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0025/)

Bayes theorem · qf-mod33-bayes-two-event-reversal · candidate





## 26. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/4, 1/2, 1/4, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/4, 1/2, 1/4, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0026/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 27. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 7/9, 1/7, 2/9, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 7/9, 1/7, 2/9, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0027/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 28. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/5, 1/2, 1/5, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/5, 1/2, 1/5, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0028/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 29. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 9/11, 1/9, 2/11, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 9/11, 1/9, 2/11, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0029/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 30. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/6, 1/2, 1/6, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/6, 1/2, 1/6, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0030/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 31. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 11/13, 1/11, 2/13, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 11/13, 1/11, 2/13, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0031/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 32. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/7, 1/2, 1/7, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/7, 1/2, 1/7, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0032/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 33. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 13/15, 1/13, 2/15, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 13/15, 1/13, 2/15, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0033/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 34. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/8, 1/2, 1/8, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/8, 1/2, 1/8, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0034/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 35. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 15/17, 1/15, 2/17, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 15/17, 1/15, 2/17, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0035/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 36. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/9, 1/2, 1/9, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/9, 1/2, 1/9, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0036/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 37. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 17/19, 1/17, 2/19, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 17/19, 1/17, 2/19, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0037/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 38. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/10, 1/2, 1/10, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/10, 1/2, 1/10, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0038/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 39. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 19/21, 1/19, 2/21, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 19/21, 1/19, 2/21, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0039/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 40. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0040/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 41. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 21/23, 1/21, 2/23, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 21/23, 1/21, 2/23, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0041/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 42. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/12, 1/2, 1/12, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/12, 1/2, 1/12, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0042/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 43. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 23/25, 1/23, 2/25, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 23/25, 1/23, 2/25, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0043/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 44. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/13, 1/2, 1/13, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/13, 1/2, 1/13, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0044/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 45. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 25/27, 1/25, 2/27, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 25/27, 1/25, 2/27, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0045/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 46. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/14, 1/2, 1/14, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/14, 1/2, 1/14, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0046/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 47. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 27/29, 1/27, 2/29, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 27/29, 1/27, 2/29, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0047/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 48. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/15, 1/2, 1/15, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/15, 1/2, 1/15, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0048/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 49. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(1, 29/31, 1/29, 2/31, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 29/31, 1/29, 2/31, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0049/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 50. Normalize Partition Hypotheses by the Same Total Evidence Mass

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

View answer and explanation(0, 1/16, 1/2, 1/16, 1/2, partition_pairwise_disjoint_exhaustive)

Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/16, 1/2, 1/16, 1/2, partition_pairwise_disjoint_exhaustive).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0050/)

Bayes theorem · qf-mod33-bayes-partition-posterior · candidate





## 51. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/4, 3/4], [1/2, 1/6], [1/8, 1/8], 1/4, 1/4, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/4, 3/4], [1/2, 1/6], [1/8, 1/8], 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0051/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 52. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/9, 7/9], [1/2, 1/7], [1/9, 1/9], 2/9, 2/9, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/9, 7/9], [1/2, 1/7], [1/9, 1/9], 2/9, 2/9, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0052/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 53. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/5, 4/5], [1/2, 1/8], [1/10, 1/10], 1/5, 1/5, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/5, 4/5], [1/2, 1/8], [1/10, 1/10], 1/5, 1/5, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0053/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 54. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/11, 9/11], [1/2, 1/9], [1/11, 1/11], 2/11, 2/11, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/11, 9/11], [1/2, 1/9], [1/11, 1/11], 2/11, 2/11, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0054/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 55. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/6, 5/6], [1/2, 1/10], [1/12, 1/12], 1/6, 1/6, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/6, 5/6], [1/2, 1/10], [1/12, 1/12], 1/6, 1/6, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0055/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 56. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/13, 11/13], [1/2, 1/11], [1/13, 1/13], 2/13, 2/13, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/13, 11/13], [1/2, 1/11], [1/13, 1/13], 2/13, 2/13, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0056/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 57. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/7, 6/7], [1/2, 1/12], [1/14, 1/14], 1/7, 1/7, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/7, 6/7], [1/2, 1/12], [1/14, 1/14], 1/7, 1/7, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0057/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 58. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/15, 13/15], [1/2, 1/13], [1/15, 1/15], 2/15, 2/15, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/15, 13/15], [1/2, 1/13], [1/15, 1/15], 2/15, 2/15, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0058/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 59. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/8, 7/8], [1/2, 1/14], [1/16, 1/16], 1/8, 1/8, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/8, 7/8], [1/2, 1/14], [1/16, 1/16], 1/8, 1/8, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0059/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 60. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/17, 15/17], [1/2, 1/15], [1/17, 1/17], 2/17, 2/17, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/17, 15/17], [1/2, 1/15], [1/17, 1/17], 2/17, 2/17, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0060/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 61. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/9, 8/9], [1/2, 1/16], [1/18, 1/18], 1/9, 1/9, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/9, 8/9], [1/2, 1/16], [1/18, 1/18], 1/9, 1/9, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0061/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 62. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/19, 17/19], [1/2, 1/17], [1/19, 1/19], 2/19, 2/19, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/19, 17/19], [1/2, 1/17], [1/19, 1/19], 2/19, 2/19, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0062/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 63. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/10, 9/10], [1/2, 1/18], [1/20, 1/20], 1/10, 1/10, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/10, 9/10], [1/2, 1/18], [1/20, 1/20], 1/10, 1/10, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0063/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 64. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/21, 19/21], [1/2, 1/19], [1/21, 1/21], 2/21, 2/21, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/21, 19/21], [1/2, 1/19], [1/21, 1/21], 2/21, 2/21, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0064/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 65. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/11, 10/11], [1/2, 1/20], [1/22, 1/22], 1/11, 1/11, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/11, 10/11], [1/2, 1/20], [1/22, 1/22], 1/11, 1/11, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0065/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 66. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/23, 21/23], [1/2, 1/21], [1/23, 1/23], 2/23, 2/23, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/23, 21/23], [1/2, 1/21], [1/23, 1/23], 2/23, 2/23, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0066/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 67. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/12, 11/12], [1/2, 1/22], [1/24, 1/24], 1/12, 1/12, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/12, 11/12], [1/2, 1/22], [1/24, 1/24], 1/12, 1/12, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0067/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 68. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/25, 23/25], [1/2, 1/23], [1/25, 1/25], 2/25, 2/25, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/25, 23/25], [1/2, 1/23], [1/25, 1/25], 2/25, 2/25, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0068/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 69. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/13, 12/13], [1/2, 1/24], [1/26, 1/26], 1/13, 1/13, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/13, 12/13], [1/2, 1/24], [1/26, 1/26], 1/13, 1/13, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0069/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 70. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/27, 25/27], [1/2, 1/25], [1/27, 1/27], 2/27, 2/27, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/27, 25/27], [1/2, 1/25], [1/27, 1/27], 2/27, 2/27, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0070/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 71. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/14, 13/14], [1/2, 1/26], [1/28, 1/28], 1/14, 1/14, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/14, 13/14], [1/2, 1/26], [1/28, 1/28], 1/14, 1/14, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0071/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 72. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/29, 27/29], [1/2, 1/27], [1/29, 1/29], 2/29, 2/29, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/29, 27/29], [1/2, 1/27], [1/29, 1/29], 2/29, 2/29, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0072/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 73. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/15, 14/15], [1/2, 1/28], [1/30, 1/30], 1/15, 1/15, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/15, 14/15], [1/2, 1/28], [1/30, 1/30], 1/15, 1/15, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0073/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 74. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([2/31, 29/31], [1/2, 1/29], [1/31, 1/31], 2/31, 2/31, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/31, 29/31], [1/2, 1/29], [1/31, 1/31], 2/31, 2/31, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0074/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 75. The Law of Total Probability Supplies the Bayes Denominator

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

View answer and explanation([1/16, 15/16], [1/2, 1/30], [1/32, 1/32], 1/16, 1/16, Yes)

Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/16, 15/16], [1/2, 1/30], [1/32, 1/32], 1/16, 1/16, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0075/)

Bayes theorem · qf-mod33-bayes-total-evidence · candidate





## 76. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0076/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 77. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0077/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 78. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0078/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 79. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0079/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 80. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0080/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 81. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0081/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 82. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0082/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 83. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0083/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 84. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0084/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 85. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0085/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 86. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0086/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 87. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0087/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 88. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0088/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 89. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0089/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 90. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0090/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 91. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0091/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 92. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0092/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 93. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0093/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 94. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0094/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 95. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0095/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 96. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0096/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 97. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0097/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 98. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0098/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 99. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0099/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate





## 100. Zero Evidence Mass Makes This Conditional Probability Undefined

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

View answer and explanation(0, undefined, No, conditional_probability_requires_positive_denominator)

Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r06-bayes-theorem-0100/)

Bayes theorem · qf-mod33-bayes-zero-denominator-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261102-r06-bayes-theorem-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261102-r06-bayes-theorem-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261102-r06-bayes-theorem-foundations/validation-report.json)