This Exam P sample reference tests Conditional Probability. The continuation event occurs unless both spouses die during the period, so its probability is 1-(0.15)(0.05)=0.9925. Dividing the husband's 0.85 survival probability by 0.9925 gives 0.856423, choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0.850 is the husband's unconditional survival probability and does not use the information that payments continue.
CThe ratio 0.95/0.9925=0.9572 instead calculates the wife's conditional survival probability under the same continuation event.
DThe value 0.9925 is the probability that payments continue, so it is the conditioning denominator rather than the requested conditional probability.
EContinued payments establish only that at least one spouse is alive; treating this as proof that the husband is alive ignores the wife-only state.
Original practice · fully worked
Original variant: redundant coolant pumps
A research freezer is kept cold by two redundant coolant pumps. Over the next service interval, pump A has failure probability 0.20 and pump B has failure probability 0.10, independently. At inspection the freezer is still being cooled, which means at least one pump is operating. Calculate the conditional probability that pump A is operating.
A 0.800
B 0.816
C 0.900
D 0.918
E 0.980
Variant answer in brief
Cooling continues with probability 1-(0.20)(0.10)=0.98. Pump A operates with probability 0.80 and that event guarantees cooling, so the conditional probability is 0.80/0.98=0.816327, choice B.
Setup
Setup
Let A and B denote the events that the corresponding pumps operate through the service interval.
Pr(A)=0.80,Pr(B)=0.90
Model
Model
Cooling is the union of the two operating events. Its complement requires both independent failures.
Pr(A∪B)=1−(0.20)(0.10)=0.98
Compute
Compute
Since an operating pump A necessarily implies cooling, normalize Pr(A) over the cooling event.
Pr(A∣A∪B)=0.980.80=0.8163265306…
Answer
Answer
The conditional probability that pump A operates is approximately 0.816.
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