Worked solution · Exam P
This Exam P sample reference tests Conditional Probability. Inclusion–exclusion gives joint working probability 0.8+0.6-0.9=0.5. Dividing by the probability that A works gives P(B works given A works)=0.5/0.8=5/8, choice C.
Independent solution
Setup
Recover the joint working probability from the two marginals and their union.
Model
Insert the supplied probabilities and then form the conditional ratio.
Compute
Normalize the joint event by the conditioning marginal.
Answer
The conditional probability is five eighths.
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Original practice · fully worked
Events A and B satisfy P(A)=0.70, P(B)=0.50, and P(exactly one of A and B)=0.40. Let I_A and I_B be their event indicators. Calculate Cov(I_A,I_B).
If x is the intersection probability, exactly one has probability 0.70+0.50-2x. Setting this to 0.40 gives x=0.40, so covariance is 0.40-(0.70)(0.50)=0.05, choice C.
Setup
Let x be the joint-event probability and express the exactly-one region through x.
Model
Solve for the joint probability and use the covariance identity for indicators.
Compute
Substitute the joint probability and two marginals.
Answer
The indicator covariance is 0.05.
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Drill the method: Exam P Conditional Probability problems
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