# Exact Event Independence, Conditional Equivalence, Complements and Zero Mass

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

batch-20261014-advance-20261027-r06-event-independence-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. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [0, 3], 1/7, 37/49, 1/49, 37/343, No)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 1/7, 37/49, 1/49, 37/343, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0001/)

Event independence · qf-mod27-eventind-audit · candidate





## 2. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 2/7, 1/7, 2/49, 2/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 1/7, 2/49, 2/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0002/)

Event independence · qf-mod27-eventind-audit · candidate





## 3. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 3/7, 1/7, 3/49, 3/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 1/7, 3/49, 3/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0003/)

Event independence · qf-mod27-eventind-audit · candidate





## 4. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 4/7, 1/7, 4/49, 4/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 1/7, 4/49, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0004/)

Event independence · qf-mod27-eventind-audit · candidate





## 5. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [0, 3], 5/7, 17/49, 5/49, 85/343, No)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 5/7, 17/49, 5/49, 85/343, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0005/)

Event independence · qf-mod27-eventind-audit · candidate





## 6. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 1/7, 2/7, 2/49, 2/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 1/7, 2/7, 2/49, 2/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0006/)

Event independence · qf-mod27-eventind-audit · candidate





## 7. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 2/7, 2/7, 4/49, 4/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 2/7, 4/49, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0007/)

Event independence · qf-mod27-eventind-audit · candidate





## 8. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 3/7, 2/7, 6/49, 6/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 2/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0008/)

Event independence · qf-mod27-eventind-audit · candidate





## 9. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [0, 3], 4/7, 23/49, 8/49, 92/343, No)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 4/7, 23/49, 8/49, 92/343, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0009/)

Event independence · qf-mod27-eventind-audit · candidate





## 10. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 5/7, 2/7, 10/49, 10/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 5/7, 2/7, 10/49, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0010/)

Event independence · qf-mod27-eventind-audit · candidate





## 11. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 1/7, 3/7, 3/49, 3/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 1/7, 3/7, 3/49, 3/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0011/)

Event independence · qf-mod27-eventind-audit · candidate





## 12. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0012/)

Event independence · qf-mod27-eventind-audit · candidate





## 13. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [0, 3], 3/7, 25/49, 9/49, 75/343, No)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 3/7, 25/49, 9/49, 75/343, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0013/)

Event independence · qf-mod27-eventind-audit · candidate





## 14. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 4/7, 3/7, 12/49, 12/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 3/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0014/)

Event independence · qf-mod27-eventind-audit · candidate





## 15. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 5/7, 3/7, 15/49, 15/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 5/7, 3/7, 15/49, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0015/)

Event independence · qf-mod27-eventind-audit · candidate





## 16. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 1/7, 4/7, 4/49, 4/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 1/7, 4/7, 4/49, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0016/)

Event independence · qf-mod27-eventind-audit · candidate





## 17. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [0, 3], 2/7, 23/49, 8/49, 46/343, No)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 2/7, 23/49, 8/49, 46/343, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0017/)

Event independence · qf-mod27-eventind-audit · candidate





## 18. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 3/7, 4/7, 12/49, 12/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 4/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0018/)

Event independence · qf-mod27-eventind-audit · candidate





## 19. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 4/7, 4/7, 16/49, 16/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 4/7, 16/49, 16/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0019/)

Event independence · qf-mod27-eventind-audit · candidate





## 20. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 5/7, 4/7, 20/49, 20/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 5/7, 4/7, 20/49, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0020/)

Event independence · qf-mod27-eventind-audit · candidate





## 21. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [0, 3], 1/7, 17/49, 5/49, 17/343, No)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 1/7, 17/49, 5/49, 17/343, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0021/)

Event independence · qf-mod27-eventind-audit · candidate





## 22. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 2/7, 5/7, 10/49, 10/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 5/7, 10/49, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0022/)

Event independence · qf-mod27-eventind-audit · candidate





## 23. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 3/7, 5/7, 15/49, 15/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 5/7, 15/49, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0023/)

Event independence · qf-mod27-eventind-audit · candidate





## 24. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [1, 3], 4/7, 5/7, 20/49, 20/49, Yes)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 5/7, 20/49, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0024/)

Event independence · qf-mod27-eventind-audit · candidate





## 25. Test Independence by the Exact Product Definition

Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

View answer and explanation([2, 3], [0, 3], 5/7, 29/49, 25/49, 145/343, No)

Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 5/7, 29/49, 25/49, 145/343, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0025/)

Event independence · qf-mod27-eventind-audit · candidate





## 26. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(1/7, 1/7, 1/49, 1/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 1/7, 1/49, 1/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0026/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 27. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(2/7, 1/7, 2/49, 2/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 1/7, 2/49, 2/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0027/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 28. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(3/7, 1/7, 3/49, 3/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 1/7, 3/49, 3/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0028/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 29. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(4/7, 1/7, 4/49, 4/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 1/7, 4/49, 4/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0029/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 30. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(5/7, 1/7, 5/49, 5/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 1/7, 5/49, 5/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0030/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 31. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(1/7, 2/7, 2/49, 1/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 2/7, 2/49, 1/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0031/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 32. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(2/7, 2/7, 4/49, 2/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 2/7, 4/49, 2/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0032/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 33. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(3/7, 2/7, 6/49, 3/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 2/7, 6/49, 3/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0033/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 34. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(4/7, 2/7, 8/49, 4/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 2/7, 8/49, 4/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0034/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 35. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(5/7, 2/7, 10/49, 5/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 2/7, 10/49, 5/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0035/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 36. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(1/7, 3/7, 3/49, 1/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 3/7, 3/49, 1/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0036/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 37. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(2/7, 3/7, 6/49, 2/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 3/7, 6/49, 2/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0037/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 38. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(3/7, 3/7, 9/49, 3/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 3/7, 9/49, 3/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0038/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 39. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(4/7, 3/7, 12/49, 4/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 3/7, 12/49, 4/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0039/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 40. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(5/7, 3/7, 15/49, 5/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 3/7, 15/49, 5/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0040/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 41. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(1/7, 4/7, 4/49, 1/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 4/7, 4/49, 1/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0041/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 42. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(2/7, 4/7, 8/49, 2/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 4/7, 8/49, 2/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0042/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 43. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(3/7, 4/7, 12/49, 3/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 4/7, 12/49, 3/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0043/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 44. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(4/7, 4/7, 16/49, 4/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 4/7, 16/49, 4/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0044/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 45. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(5/7, 4/7, 20/49, 5/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 4/7, 20/49, 5/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0045/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 46. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(1/7, 5/7, 5/49, 1/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 5/7, 5/49, 1/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0046/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 47. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(2/7, 5/7, 10/49, 2/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 5/7, 10/49, 2/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0047/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 48. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(3/7, 5/7, 15/49, 3/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 5/7, 15/49, 3/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0048/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 49. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(4/7, 5/7, 20/49, 4/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 5/7, 20/49, 4/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0049/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 50. Use Conditional Equivalence Only with Positive Denominator

Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).

View answer and explanation(5/7, 5/7, 25/49, 5/7, Yes, Yes)

Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 5/7, 25/49, 5/7, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0050/)

Event independence · qf-mod27-eventind-conditional-equivalence · candidate





## 51. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 6/7, 6/49, 6/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0051/)

Event independence · qf-mod27-eventind-complement · candidate





## 52. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 5/7, 5/49, 5/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 5/49, 5/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0052/)

Event independence · qf-mod27-eventind-complement · candidate





## 53. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 4/7, 4/49, 4/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 4/49, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0053/)

Event independence · qf-mod27-eventind-complement · candidate





## 54. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 3/7, 3/49, 3/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 3/49, 3/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0054/)

Event independence · qf-mod27-eventind-complement · candidate





## 55. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 2/7, 2/49, 2/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 2/49, 2/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0055/)

Event independence · qf-mod27-eventind-complement · candidate





## 56. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 6/7, 12/49, 12/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0056/)

Event independence · qf-mod27-eventind-complement · candidate





## 57. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 5/7, 10/49, 10/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 10/49, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0057/)

Event independence · qf-mod27-eventind-complement · candidate





## 58. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 4/7, 8/49, 8/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 8/49, 8/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0058/)

Event independence · qf-mod27-eventind-complement · candidate





## 59. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 3/7, 6/49, 6/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0059/)

Event independence · qf-mod27-eventind-complement · candidate





## 60. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 2/7, 4/49, 4/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 4/49, 4/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0060/)

Event independence · qf-mod27-eventind-complement · candidate





## 61. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 6/7, 18/49, 18/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 18/49, 18/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0061/)

Event independence · qf-mod27-eventind-complement · candidate





## 62. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 5/7, 15/49, 15/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 15/49, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0062/)

Event independence · qf-mod27-eventind-complement · candidate





## 63. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 4/7, 12/49, 12/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0063/)

Event independence · qf-mod27-eventind-complement · candidate





## 64. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 3/7, 9/49, 9/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 9/49, 9/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0064/)

Event independence · qf-mod27-eventind-complement · candidate





## 65. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 2/7, 6/49, 6/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 6/49, 6/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0065/)

Event independence · qf-mod27-eventind-complement · candidate





## 66. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 6/7, 24/49, 24/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 24/49, 24/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0066/)

Event independence · qf-mod27-eventind-complement · candidate





## 67. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 5/7, 20/49, 20/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 20/49, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0067/)

Event independence · qf-mod27-eventind-complement · candidate





## 68. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 4/7, 16/49, 16/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 16/49, 16/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0068/)

Event independence · qf-mod27-eventind-complement · candidate





## 69. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 3/7, 12/49, 12/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 12/49, 12/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0069/)

Event independence · qf-mod27-eventind-complement · candidate





## 70. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 2/7, 8/49, 8/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 8/49, 8/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0070/)

Event independence · qf-mod27-eventind-complement · candidate





## 71. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 6/7, 30/49, 30/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 30/49, 30/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0071/)

Event independence · qf-mod27-eventind-complement · candidate





## 72. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 5/7, 25/49, 25/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 25/49, 25/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0072/)

Event independence · qf-mod27-eventind-complement · candidate





## 73. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 4/7, 20/49, 20/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 20/49, 20/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0073/)

Event independence · qf-mod27-eventind-complement · candidate





## 74. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 3/7, 15/49, 15/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 15/49, 15/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0074/)

Event independence · qf-mod27-eventind-complement · candidate





## 75. Audit Complement Independence without Assuming the Original Flag

Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

View answer and explanation(Yes, [0, 1], 2/7, 10/49, 10/49, Yes)

Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 10/49, 10/49, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0075/)

Event independence · qf-mod27-eventind-complement · candidate





## 76. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0076/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 77. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0077/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 78. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0078/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 79. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0079/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 80. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0080/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 81. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0081/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 82. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0082/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 83. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0083/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 84. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0084/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 85. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0085/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 86. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0086/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 87. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0087/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 88. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0088/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 89. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0089/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 90. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0090/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 91. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0091/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 92. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0092/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 93. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0093/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 94. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0094/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 95. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0095/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 96. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0096/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 97. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0097/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 98. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0098/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 99. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0099/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate





## 100. A Zero-Mass Conditioning Event Still Has a Product Definition

Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

View answer and explanation(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r06-event-independence-0100/)

Event independence · qf-mod27-eventind-zero-probability-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261027-r06-event-independence-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261027-r06-event-independence-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261027-r06-event-independence-foundations/validation-report.json)