Worked solution · Exam P
This Exam P sample reference tests Binomial Distribution. The twenty-year count is binomial. Summing its probabilities for zero, one, and two events gives 0.92452, choice E.
Independent solution
Setup
At most one event can occur in each year, independently, so the twenty-year count is binomial with event probability 0.05.
Model
Fewer than three events includes the three disjoint counts zero, one, and two.
Compute
Adding those three binomial masses gives approximately 0.924516.
Answer
The probability of fewer than three events is approximately 0.92, selecting choice E.
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.
Original practice · fully worked
During each of fifteen independent service periods, an outage occurs with probability 0.08 and at most one outage can occur per period. Given that the total number of outages is no more than one, find the probability that it is exactly one.
The ratio of the one-outage mass to the zero-outage mass is 15(0.08)/0.92. Normalizing those two masses gives 0.5660, choice D.
Setup
The total outage count across 15 independent periods is binomial with outage probability 0.08.
Model
Under the condition of at most one outage, only the zero- and one-outage masses remain. Their ratio simplifies before normalization.
Compute
Normalizing the two admissible masses gives an exactly-one probability of approximately 0.566038.
Answer
The conditional probability of exactly one outage is approximately 0.5660, selecting choice D.
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