Worked solution · Exam P
This Exam P sample reference tests Inclusion-Exclusion. The two event probabilities are 0.30 and 0.15, while their union has probability 0.39. Inclusion-exclusion therefore gives intersection probability 0.30+0.15-0.39=0.060, choice B.
Independent solution
Setup
Represent the two types of utilization by events E and H and complement the given zero-use probabilities.
Model
The probability of at least one of the two utilization types is the complement of having neither type.
Compute
Rearrange the two-event inclusion-exclusion identity to isolate the overlap.
Answer
The probability of experiencing both utilization types is 0.060.
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Original practice · fully worked
During equipment inspections, 42% trigger an acoustic monitor and 30% trigger a thermal monitor. Among inspections that trigger the acoustic monitor, 25% also trigger the thermal monitor. Determine the probability that a randomly chosen inspection triggers neither monitor.
The joint trigger probability is 0.42(0.25)=0.105. Thus the probability of at least one trigger is 0.42+0.30-0.105=0.615, leaving 0.385 for neither, choice C.
Setup
Let A and T denote acoustic and thermal triggers.
Model
First recover the joint trigger probability from the conditional probability.
Compute
Find the union and then take its complement.
Answer
The probability that neither monitor triggers is 0.385.
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