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Exact Priors, Likelihoods, Partition Normalization and Zero Denominators

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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