This Exam P sample reference tests Conditional Probability. Given the husband survives, the wife's death probability is 0.01/(0.96+0.01)=1/97. Expected claims are 10,000/97, leaving expected premium excess about 897, choice E.
How to solve this Conditional Probability question
Setup
Setup
Condition on the husband being alive. The conditioning probability includes the states in which both spouses survive and in which only the husband survives.
P(H)=0.96+0.01=0.97
P(Wc∩H)=0.01
Model
Model
Within that conditioning event, the claim is paid only in the state where the wife dies. Divide that joint-state probability by the husband-survival probability.
P(Wc∣H)=0.970.01=971
Compute
Compute
Multiplying the conditional claim probability by 10,000 gives expected claim 103.09. Subtracting it from the premium leaves approximately 896.91.
E[claim∣H]=10000971=103.0928
E[premium excess∣H]=1000−103.0928=896.9072
Answer
Answer
The conditional expected premium excess rounds to 897, selecting choice E.
897(E)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A uses an unconditional wife-death probability instead of restricting to states where the husband survives.
BChoice B conditions on a different survivor event.
CChoice C subtracts a claim amount weighted by an incorrect conditional probability.
DChoice D uses the raw joint probability 0.01 without dividing by the husband-survival probability 0.97.
Original practice · fully worked
Original variant: conditional warranty margin after one component survives
A paired-device warranty collects 60 dollars total and pays 300 dollars if either device fails during a year. The probabilities that both survive, only device A survives, and only device B survives are 0.90, 0.04, and 0.03. Given that A survives, find expected premium minus payout.
A 27.45 dollars
C 40.65 dollars
B 47.23 dollars
D 50.32 dollars
E 60.00 dollars
Variant answer in brief
Given A survives, B fails with probability 0.04/0.94. The expected payout is 12.766, leaving a margin of 47.234 dollars.
Setup
Setup
Given that device A survives, the admissible states are both devices surviving and only A surviving. Their total probability is 0.94.
P(A survives)=0.90+0.04=0.94
Model
Model
A payout occurs within this conditioning event only when B fails, whose joint-state probability is 0.04.
P(B fails∣A survives)=0.940.04
Compute
Compute
The conditional expected payout is approximately 12.77 dollars. Subtracting it from the 60-dollar premium gives approximately 47.23 dollars.
E[C∣A]=3000.940.04=12.7660
60−12.7660=47.2340
Answer
Answer
The conditional expected margin is approximately 47.23 dollars, selecting choice B.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
