This Exam P sample reference tests Conditional Survival. Condition on survival to age 40, then evaluate death before age 50 from the supplied lifetime CDF. The probability is about 0.0696, producing expected benefit 348 and choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 333 is consistent with using an unconditional interval probability rather than conditioning on survival to age 40.
Original practice · fully worked
Original variant: conditional benefit under exponential mortality
A component already operating for 5 years has an exponential remaining-life model with mean 20 years. A warranty pays 2,000 dollars if it fails before year 12. Find the expected payment conditional on operation through year 5.
A 393.47 dollars
B 590.62 dollars
C 704.69 dollars
D 786.94 dollars
E 1,409.38 dollars
Variant answer in brief
Memorylessness leaves a seven-year horizon. The conditional failure probability is 1-exp(-7/20), so expected payment is about 590.6 dollars.
Setup
Setup
Conditional on operation through year 5, failure before year 12 means failure during the next seven years.
P(T<12∣T>5)=1−e−(12−5)/20
Model
Model
Memorylessness leaves an exponential seven-year failure probability, which is multiplied by the 2000-dollar warranty amount.
E[B∣T>5]=2000(1−e−7/20)
Compute
Compute
The seven-year failure probability is 1 minus exp(-7/20), yielding expected payment 590.624 dollars.
E[B∣T>5]=590.624
Answer
Answer
The conditional expected warranty payment is 590.62 dollars, corresponding to choice B.
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