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Multiplication law; applicant passing test

Short answer

For any events A and B, P(A ∩ B) = P(A) P(B|A). Using total probability, P(pass) = 0.7×0.9 + 0.3×0.4 = 0.75. By Bayes, P(capable | pass) = 0.63 / 0.75 = 0.84.

State the law of multiplication of probability. An Electronics company has an engineering position open. The Probability that an applicant is capable is 0.7. Each applicant is given written test and oral examination. A capable applicant passes with Probability 0.9 while an incapable applicant passes with Probability of 0.4. Find (a) the probability that an applicant passes the test (b) the probability that the applicant is capable given he/she passes the test.

Law of multiplication of probability

For two events A and B, the probability that both occur is

P(A ∩ B) = P(A) · P(B | A) = P(B) · P(A | B)

If A and B are independent, P(B | A) = P(B), so P(A ∩ B) = P(A) · P(B). For three events, P(A ∩ B ∩ C) = P(A) P(B|A) P(C|A ∩ B).

Applicant problem

Let C be "applicant is capable" and T be "applicant passes".

P(C) = 0.7,   P(C′) = 0.3
P(T|C) = 0.9, P(T|C′) = 0.4

(a) Probability that an applicant passes

P(T) = P(C) P(T|C) + P(C′) P(T|C′)
     = (0.7)(0.9) + (0.3)(0.4)
     = 0.63 + 0.12
     = 0.75

(b) Probability of being capable given the applicant passes

P(C|T) = P(C) P(T|C) / P(T) = 0.63 / 0.75 = 0.84

Answer: (a) 0.75, (b) 0.84.

✓ Answer checked by Claude Sonnet 5.5

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