Worked solution · Exam P
This Exam P sample reference tests Expected Value. Death in year k requires survival through the preceding k-1 years, so its probability is 0.99⁽ᵏ⁻¹⁾ times 0.01. Weighting the five possible benefits by those probabilities gives 740.0747, which corresponds to choice B.
Independent solution
Setup
Index the possible payment year by k from 1 through 5. Reaching year k alive requires k-1 survivals.
Model
The benefit declines by 5,000 for each later payment year.
Compute
Sum benefit times probability over the five mutually exclusive death years.
Answer
The expected benefit rounds to 740.
Test the method on unseen problems
30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
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Original practice · fully worked
A research drone attempts three navigation checkpoints in sequence. Conditional on reaching a checkpoint, its probabilities of completing checkpoints 1, 2, and 3 are 0.80, 0.75, and 0.50. It earns 100 units immediately after each completed checkpoint and an additional 300 units if it completes all three. Calculate the expected total reward.
The cumulative checkpoint-completion probabilities are 0.80, 0.60, and 0.30. The three checkpoint rewards contribute 170 in expectation and the completion bonus contributes 90, for a total of 260 and choice D.
Setup
Convert the conditional success rates into cumulative probabilities of completing each checkpoint.
Model
Use one indicator for each completed checkpoint. The bonus uses the third indicator because checkpoint 3 can be completed only after the first two.
Compute
Apply linearity of expectation to the accumulated checkpoint rewards and the terminal bonus.
Answer
The expected total reward is 260 units.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 387 Exam P univariate random variables practice problems.
Drill the method: Exam P Tail Integral Formula and Expected Value problems
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