Worked solution · Exam P
This Exam P sample reference tests Inclusion–Exclusion. Writing the therapist probability as t and the chiropractor probability as t+0.14 makes the union equation linear; t=0.48, choice D.
Independent solution
Setup
Let t=Pr(T). The chiropractor probability is then t+0.14, and the probability of at least one treatment is 1-0.12=0.88.
Model
Apply the two-event addition rule, subtracting the 0.22 overlap once.
Compute
Substituting Pr(C)=t+0.14 yields 0.88=2t-0.08, so 2t=0.96 and t=0.48.
Answer
The therapist probability is 0.48, corresponding to choice D.
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Original practice · fully worked
Two data-quality checks fail with probabilities 0.52 and 0.41. The probability that exactly one of the checks fails is 0.47. Determine the probability that both checks fail.
Exactly-one probability equals the marginal sum minus twice the overlap. Solving 0.47=0.93-2x gives x=0.23, choice B.
Setup
Let x be the probability that both checks fail.
Model
The exactly-one event consists of A without B and B without A, so its probability is the sum of the two marginals minus twice their overlap.
Compute
Substitution gives 0.47=0.93-2x, from which x=0.23.
Answer
Therefore both checks fail with probability 0.23, which is choice B.
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