Worked solution · Exam P
This Exam P sample reference tests Set Probability. This is a three-set incidence calculation under an exactly-two-or-zero participation rule. Writing the three pair-only proportions as unknowns turns the supplied marginal rates into a linear system; the life-and-retirement cell is 0.125, so choice A is correct.
Independent solution
Setup
Because a participating employee selects exactly two plans, partition participants into the three possible plan pairs.
Model
Each marginal participation rate is the sum of the two pair cells containing that plan.
Compute
Add the life and retirement equations, then subtract the health equation to isolate twice the requested pair.
Answer
The requested joint participation rate is 12.5%.
Test the method on unseen problems
30 original general probability problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
Error diagnosis
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Original practice · fully worked
A production batch may receive any of three review tags: thermal, vibration, and alignment. No batch receives all three tags. Historical rates are 0.55 for thermal, 0.50 for vibration, and 0.45 for alignment, while 0.90 of batches receive at least one tag. Calculate the probability that a batch receives exactly two tags.
Let N count tags on a batch. Its mean is the sum of the three marginal rates, 1.50; because N can only be zero, one, or two, the tail-sum identity gives 1.50=P(N at least 1)+P(N at least 2)=0.90+P(N=2). Hence the answer is 0.60, choice C.
Setup
Let N be the number of tags attached to one batch and add the three indicator means.
Model
The prohibition on all three tags makes two the largest possible count. Apply the tail-sum identity for a bounded nonnegative integer variable.
Compute
Insert the probability of at least one tag and isolate the probability of two tags.
Answer
A batch receives exactly two tags with probability 0.60.
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