Worked solution · Exam P
This Exam P sample reference tests Binomial Tail Probability. With three independent components and failure probability 0.05, overheating means two or three failures. The binomial tail is 3(0.05²)(0.95)+0.05³=0.00725, choice A.
Independent solution
Setup
Let the number of failed components be binomial with three trials and failure probability 0.05.
Model
Overheating requires at least two failures, so include the disjoint cases of exactly two and exactly three failures.
Compute
The two probabilities are 0.007125 and 0.000125. Their sum is 0.00725.
Answer
The overheat probability rounds to 0.007.
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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 controller has four independent modules, each failing with probability 0.10. The controller shuts down if at least three modules fail. Find the shutdown probability.
The probabilities of exactly three and exactly four failures are 4(0.1³)(0.9)=0.0036 and 0.0001. Their sum is 0.0037, choice C.
Setup
Let the number of failed modules be binomial with four trials and failure probability 0.10.
Model
Shutdown requires exactly three or exactly four failures.
Compute
The two probabilities are 0.0036 and 0.0001, giving a shutdown probability of 0.0037.
Answer
The shutdown probability is 0.0037.
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