Worked solution · Exam P
This Exam P sample reference tests Negative Binomial Distribution. The first nine trials must contain exactly two successes, and the tenth trial must succeed. This has probability C(9,2)(0.45)³(0.55)⁷=0.049943, so choice B is correct.
Independent solution
Setup
For the third success to occur at trial ten, exactly two of the preceding nine trials must succeed.
Model
Choose the locations of the first two successes and multiply by the independent trial probabilities.
Compute
Insert the stated success probability.
Answer
Rounded to the precision of the choices, the probability is 0.0499.
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.
Error diagnosis
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Original practice · fully worked
A network monitor classifies each arriving packet independently. A packet is rejected with probability 0.30, and monitoring stops as soon as the third rejected packet is observed. Calculate the expected total number of packets observed.
The stopping time is negative binomial with three target events and target probability 0.30. Its mean is 3/0.30=10 packets, so choice D is correct.
Setup
Treat a rejection as the target event and let N be the trial on which the third rejection occurs.
Model
The expected trial count for the rth target event in independent Bernoulli trials is r divided by the target probability.
Compute
Substitute the rejection count and rejection probability.
Answer
The monitor observes ten packets on average.
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