Exact Event Independence, Conditional Equivalence, Complements and Zero Mass
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Event independence · qf-mod27-eventind-zero-probability-boundary · candidate
Canonical question JSONL · English JSONL · Validation report