Worked solution · Exam P
This Exam P sample reference tests Joint Distributions. This problem complements the three joint cells whose combined count is below two. Those cells have total mass 21/54, leaving 33/54=0.611111 and choice E.
Independent solution
Setup
Let S=X+Y be the combined count. It is shorter to complement the outcomes with totals zero or one.
Model
Only three support points have a combined count below two.
Compute
Evaluate the joint mass at those points and subtract their sum from one.
Answer
The probability of a combined count of at least two rounds to 0.61.
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30 original multivariate 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 machine is currently in operating mode A, B, or C with probabilities 0.20, 0.50, and 0.30. At the next inspection, the probabilities that it remains in its current mode are 0.70, 0.60, and 0.80 for modes A, B, and C, respectively. Calculate the probability that its mode changes.
The mode-specific change probabilities are 0.30, 0.40, and 0.20. Weighting them by the current-mode probabilities gives 0.20(0.30)+0.50(0.40)+0.30(0.20)=0.32 and choice C.
Setup
Take the complement of each mode-specific stay probability.
Model
The current modes form a partition, so apply the law of total probability to the change event M.
Compute
Weight the three conditional change probabilities by their current-mode probabilities.
Answer
The machine changes mode with probability 0.32.
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