# Declared Event Families and Typed Occurrences

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

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

batch-20261009-advance-20261013-r06-event-foundations · 2026-10-09 · 100 questions

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



## 1. Declared Event and Relative Complement Partition

S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([], [0], 1)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [0], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0001/)

Event · qf-density13-event-complement-partition · candidate





## 2. Declared Event and Relative Complement Partition

S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[0]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0], [], 1)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0002/)

Event · qf-density13-event-complement-partition · candidate





## 3. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0], [1, 2, 3, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [1, 2, 3, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0003/)

Event · qf-density13-event-complement-partition · candidate





## 4. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([1], [0, 2, 3, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1], [0, 2, 3, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0004/)

Event · qf-density13-event-complement-partition · candidate





## 5. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 1], [2, 3, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0005/)

Event · qf-density13-event-complement-partition · candidate





## 6. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([2], [0, 1, 3, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2], [0, 1, 3, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0006/)

Event · qf-density13-event-complement-partition · candidate





## 7. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 2], [1, 3, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0007/)

Event · qf-density13-event-complement-partition · candidate





## 8. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([1, 2], [0, 3, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0008/)

Event · qf-density13-event-complement-partition · candidate





## 9. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 1, 2], [3, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2], [3, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0009/)

Event · qf-density13-event-complement-partition · candidate





## 10. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([3], [0, 1, 2, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([3], [0, 1, 2, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0010/)

Event · qf-density13-event-complement-partition · candidate





## 11. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 3], [1, 2, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0011/)

Event · qf-density13-event-complement-partition · candidate





## 12. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([1, 3], [0, 2, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0012/)

Event · qf-density13-event-complement-partition · candidate





## 13. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 1, 3], [2, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 3], [2, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0013/)

Event · qf-density13-event-complement-partition · candidate





## 14. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2, 3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([2, 3], [0, 1, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0014/)

Event · qf-density13-event-complement-partition · candidate





## 15. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2, 3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 2, 3], [1, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 3], [1, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0015/)

Event · qf-density13-event-complement-partition · candidate





## 16. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2, 3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([1, 2, 3], [0, 4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 3], [0, 4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0016/)

Event · qf-density13-event-complement-partition · candidate





## 17. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2, 3]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 1, 2, 3], [4], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [4], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0017/)

Event · qf-density13-event-complement-partition · candidate





## 18. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([4], [0, 1, 2, 3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([4], [0, 1, 2, 3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0018/)

Event · qf-density13-event-complement-partition · candidate





## 19. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 4], [1, 2, 3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0019/)

Event · qf-density13-event-complement-partition · candidate





## 20. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([1, 4], [0, 2, 3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0020/)

Event · qf-density13-event-complement-partition · candidate





## 21. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 1, 4], [2, 3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 4], [2, 3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0021/)

Event · qf-density13-event-complement-partition · candidate





## 22. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2, 4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([2, 4], [0, 1, 3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0022/)

Event · qf-density13-event-complement-partition · candidate





## 23. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2, 4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 2, 4], [1, 3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 4], [1, 3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0023/)

Event · qf-density13-event-complement-partition · candidate





## 24. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2, 4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([1, 2, 4], [0, 3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 4], [0, 3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0024/)

Event · qf-density13-event-complement-partition · candidate





## 25. Declared Event and Relative Complement Partition

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2, 4]⊆S. Give (increasing E, increasing S\E, n).

View answer and explanation([0, 1, 2, 4], [3], 5)

Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [3], 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0025/)

Event · qf-density13-event-complement-partition · candidate





## 26. Event Union, Intersection, and Outside Complement

S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). A=[], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([], [], [0])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0026/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 27. Event Union, Intersection, and Outside Complement

S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). A=[0], B=[0] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0], [0], [])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [0], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0027/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 28. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0], [], [1, 2, 3, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [], [1, 2, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0028/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 29. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([1], [1], [0, 2, 3, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1], [1], [0, 2, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0029/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 30. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 1], [], [2, 3, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1], [], [2, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0030/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 31. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2], B=[2] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([2], [2], [0, 1, 3, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2], [2], [0, 1, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0031/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 32. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 2], [], [1, 3, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2], [], [1, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0032/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 33. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([1, 2], [1], [0, 3, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2], [1], [0, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0033/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 34. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 1, 2], [], [3, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2], [], [3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0034/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 35. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[3], B=[3] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([3], [3], [0, 1, 2, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([3], [3], [0, 1, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0035/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 36. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 3], [], [1, 2, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 3], [], [1, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0036/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 37. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 3], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([1, 3], [1], [0, 2, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 3], [1], [0, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0037/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 38. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 1, 3], [], [2, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 3], [], [2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0038/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 39. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2, 3], B=[2] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([2, 3], [2], [0, 1, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 3], [2], [0, 1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0039/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 40. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 2, 3], [], [1, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 3], [], [1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0040/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 41. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2, 3], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([1, 2, 3], [1], [0, 4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 3], [1], [0, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0041/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 42. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 1, 2, 3], [], [4])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [], [4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0042/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 43. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[4], B=[4] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([4], [4], [0, 1, 2, 3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([4], [4], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0043/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 44. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 4], [], [1, 2, 3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 4], [], [1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0044/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 45. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 4], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([1, 4], [1], [0, 2, 3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 4], [1], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0045/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 46. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 1, 4], [], [2, 3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 4], [], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0046/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 47. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2, 4], B=[2] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([2, 4], [2], [0, 1, 3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 4], [2], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0047/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 48. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 2, 4], [], [1, 3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 4], [], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0048/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 49. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2, 4], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([1, 2, 4], [1], [0, 3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 4], [1], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0049/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 50. Event Union, Intersection, and Outside Complement

S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.

View answer and explanation([0, 1, 2, 4], [], [3])

Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [], [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0050/)

Event · qf-density13-event-boolean-algebra-pair · candidate





## 51. Partition-Union Event Eligibility and Least Containing Event

S=Fin(1)={0,…,0}; F consists of unions of complete disjoint nonempty blocks=[[0]]. Candidate E=[]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(Yes, [], [])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (Yes, [], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0051/)

Event · qf-density13-event-partition-family-qualification · candidate





## 52. Partition-Union Event Eligibility and Least Containing Event

S=Fin(3)={0,…,2}; F consists of unions of complete disjoint nonempty blocks=[[0, 1], [2]]. Candidate E=[0, 1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(Yes, [], [0, 1])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (Yes, [], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0052/)

Event · qf-density13-event-partition-family-qualification · candidate





## 53. Partition-Union Event Eligibility and Least Containing Event

S=Fin(3)={0,…,2}; F consists of unions of complete disjoint nonempty blocks=[[0, 1], [2]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 1]], [0, 1])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1]], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0053/)

Event · qf-density13-event-partition-family-qualification · candidate





## 54. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2], [1], [3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 2]], [0, 2])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2]], [0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0054/)

Event · qf-density13-event-partition-family-qualification · candidate





## 55. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2], [0], [3], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[1, 2]], [1, 2])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2]], [1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0055/)

Event · qf-density13-event-partition-family-qualification · candidate





## 56. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2], [3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 1, 2]], [0, 1, 2])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2]], [0, 1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0056/)

Event · qf-density13-event-partition-family-qualification · candidate





## 57. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3], [1], [2], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 3]], [0, 3])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 3]], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0057/)

Event · qf-density13-event-partition-family-qualification · candidate





## 58. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 3], [0], [2], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[1, 3]], [1, 3])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 3]], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0058/)

Event · qf-density13-event-partition-family-qualification · candidate





## 59. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 3], [2], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 1, 3]], [0, 1, 3])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 3]], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0059/)

Event · qf-density13-event-partition-family-qualification · candidate





## 60. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 3], [0], [1], [4], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[2, 3]], [2, 3])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 3]], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0060/)

Event · qf-density13-event-partition-family-qualification · candidate





## 61. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2, 3], [1], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 2, 3]], [0, 2, 3])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2, 3]], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0061/)

Event · qf-density13-event-partition-family-qualification · candidate





## 62. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2, 3], [0], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[1, 2, 3]], [1, 2, 3])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2, 3]], [1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0062/)

Event · qf-density13-event-partition-family-qualification · candidate





## 63. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2, 3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 1, 2, 3]], [0, 1, 2, 3])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2, 3]], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0063/)

Event · qf-density13-event-partition-family-qualification · candidate





## 64. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 4], [1], [2], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 4]], [0, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 4]], [0, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0064/)

Event · qf-density13-event-partition-family-qualification · candidate





## 65. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 4], [0], [2], [3], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[1, 4]], [1, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 4]], [1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0065/)

Event · qf-density13-event-partition-family-qualification · candidate





## 66. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 4], [2], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 1, 4]], [0, 1, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 4]], [0, 1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0066/)

Event · qf-density13-event-partition-family-qualification · candidate





## 67. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 4], [0], [1], [3], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[2, 4]], [2, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 4]], [2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0067/)

Event · qf-density13-event-partition-family-qualification · candidate





## 68. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2, 4], [1], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 2, 4]], [0, 2, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2, 4]], [0, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0068/)

Event · qf-density13-event-partition-family-qualification · candidate





## 69. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2, 4], [0], [3], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[1, 2, 4]], [1, 2, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2, 4]], [1, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0069/)

Event · qf-density13-event-partition-family-qualification · candidate





## 70. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2, 4], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 1, 2, 4]], [0, 1, 2, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2, 4]], [0, 1, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0070/)

Event · qf-density13-event-partition-family-qualification · candidate





## 71. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[3, 4], [0], [1], [2], [5], [6], [7]]. Candidate E=[3]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[3, 4]], [3, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[3, 4]], [3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0071/)

Event · qf-density13-event-partition-family-qualification · candidate





## 72. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 3, 4]], [0, 3, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 3, 4]], [0, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0072/)

Event · qf-density13-event-partition-family-qualification · candidate





## 73. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 3, 4], [0], [2], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[1, 3, 4]], [1, 3, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 3, 4]], [1, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0073/)

Event · qf-density13-event-partition-family-qualification · candidate





## 74. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 3, 4], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[0, 1, 3, 4]], [0, 1, 3, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 3, 4]], [0, 1, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0074/)

Event · qf-density13-event-partition-family-qualification · candidate





## 75. Partition-Union Event Eligibility and Least Containing Event

S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 3, 4], [0], [1], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.

View answer and explanation(No, [[2, 3, 4]], [2, 3, 4])

Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 3, 4]], [2, 3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0075/)

Event · qf-density13-event-partition-family-qualification · candidate





## 76. Typed Observations, Event Occurrence, and First Hit

S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[], ordered outcomes=[]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([], [], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0076/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 77. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, No, No], [invalid-outcome, invalid-outcome, invalid-outcome], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, No], [invalid-outcome, invalid-outcome, invalid-outcome], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0077/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 78. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, Yes], [False, False, False], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, False], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0078/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 79. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, Yes], [True, True, True], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, True], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0079/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 80. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, No, No], [False, invalid-outcome, invalid-outcome], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, No], [False, invalid-outcome, invalid-outcome], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0080/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 81. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, No, No], [True, invalid-outcome, invalid-outcome], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, No], [True, invalid-outcome, invalid-outcome], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0081/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 82. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, Yes, No], [invalid-outcome, False, invalid-outcome], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, No], [invalid-outcome, False, invalid-outcome], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0082/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 83. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, No], [False, False, invalid-outcome], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, False, invalid-outcome], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0083/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 84. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, No], [True, False, invalid-outcome], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, False, invalid-outcome], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0084/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 85. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, Yes, No], [invalid-outcome, True, invalid-outcome], 1)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, No], [invalid-outcome, True, invalid-outcome], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0085/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 86. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, No], [False, True, invalid-outcome], 1)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, True, invalid-outcome], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0086/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 87. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, No], [True, True, invalid-outcome], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, True, invalid-outcome], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0087/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 88. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, No, Yes], [invalid-outcome, invalid-outcome, False], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, Yes], [invalid-outcome, invalid-outcome, False], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0088/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 89. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, No, Yes], [False, invalid-outcome, False], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, invalid-outcome, False], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0089/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 90. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, No, Yes], [True, invalid-outcome, False], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, invalid-outcome, False], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0090/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 91. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, Yes, Yes], [invalid-outcome, False, False], none)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, False, False], none).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0091/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 92. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, Yes], [True, False, False], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, False, False], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0092/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 93. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, Yes, Yes], [invalid-outcome, True, False], 1)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, True, False], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0093/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 94. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, Yes], [False, True, False], 1)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, True, False], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0094/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 95. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, Yes], [True, True, False], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, False], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0095/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 96. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, No, Yes], [invalid-outcome, invalid-outcome, True], 2)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, Yes], [invalid-outcome, invalid-outcome, True], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0096/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 97. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, No, Yes], [False, invalid-outcome, True], 2)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, invalid-outcome, True], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0097/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 98. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, No, Yes], [True, invalid-outcome, True], 0)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, invalid-outcome, True], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0098/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 99. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([No, Yes, Yes], [invalid-outcome, False, True], 2)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, False, True], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0099/)

Event · qf-density13-event-occurrence-first-hit · candidate





## 100. Typed Observations, Event Occurrence, and First Hit

S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).

View answer and explanation([Yes, Yes, Yes], [False, False, True], 2)

Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, True], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r06-event-0100/)

Event · qf-density13-event-occurrence-first-hit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r06-event-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261009-advance-20261013-r06-event-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r06-event-foundations/validation-report.json)