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The health department of a city routinely conducts two independent inspections
of each restaurant, with the restaurant passing only if both inspectors pass it.
Inspector A is very experienced and hence passes only 2 percent of restaurants
that actually do have health code violations. Inspector B is less experienced
and pass 7 percent of restaurants with health code violations what is the
probability that
(a) Inspector A passes a restaurant, give that inspector B has found a violation?
(b) Inspector B passes a restaurant with a violation given that inspector A passes it?
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This problem is on conditional probability.
(a) This question is to find the conditional probability
P(A passes AND B found a violation)
P(A passes | B found a violation) = --------------------------------------
P(B found a violation)
These events, (A passes) and (B found a violation) are independent, so the probability
of their intersection is the product of probabilities
P(A passes AND B found a violation) = P(A passes) * P(B found a violation) = 0.02*(1-0.07) =
= 0.02*0.93.
So, the conditional probability is P(A passes | B found a violation)) = = 0.02.
This is the ANSWER to question (a).
(b) For question (b), the logic/(the reasoning) is similar, and it leads to the solution
P(B passes AND A passes) 0.07*0.02
P(B passes | A found a violation) = -------------------------- = --------------- = 0.07.
P(A passes) 0.02
This is the ANSWER to question (b).
The answers are consistent with the intuitive feeling / understanding of independent events.
Solved.