Worked solution · Exam P
This Exam P sample reference tests Mutually Exclusive Events. A union probability equals the sum of three event probabilities only when the selected events are pairwise disjoint. A, B, and E cannot occur in any pair, whereas every other listed triple contains an overlapping pair, so choice A is correct.
Independent solution
Setup
Use the additivity criterion for a finite collection of events.
Model
Check the proposed events A, B, and E two at a time using their year-specific requirements.
Compute
Each intersection is impossible: A excludes any second- or third-year theft, while B and E impose contradictory second-year states.
Answer
The additive triple is A, B, and E.
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Error diagnosis
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Original practice · fully worked
A fair six-sided selector outputs one of the integers 1 through 6. Define U={1,2,3}, V={3,4}, and W={4,5}. Calculate [P(U)+P(V)+P(W)]-P(U union V union W).
The three individual probabilities sum to 7/6, while their union contains five of the six outputs and has probability 5/6. The additivity gap is 2/6=1/3, choice B.
Setup
Compute the three event probabilities from their numbers of selector outcomes.
Model
Identify the distinct outputs included in at least one event.
Compute
Subtract the union probability from the sum of the marginals.
Answer
The overlap creates an additivity gap of one third.
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