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Conditional probability; voltage regulator suppliers

Short answer

(i) P(spec) = 0.60×0.95 + 0.30×0.80 + 0.10×0.65 = 0.875. (ii) P(supplier B | spec) = 0.240 / 0.875 ≈ 0.2743.

Define conditional probability. An assembly plant receives its voltage regulators from these three different suppliers, 60% from supplier A, 30% from supplier B, and 10% from supplier C. It is also known that 95% of voltage regulators from A, 80% of these from B, and 65% these from C perform according to specifications. What is the probability that i) Any one voltage regulator received by the plant will perform according to specifications ii) A voltage regulator that perform according to specification came from B

Conditional probability

The conditional probability of event B given that event A has occurred (P(A) > 0) is

P(B | A) = P(A ∩ B) / P(A)

It is the probability of B when the sample space is reduced to A. Example: a card drawn from a pack is known to be a face card; the probability that it is a king is 4/12 = 1/3.

Voltage regulators

Let A, B, C be the suppliers and S the event that a regulator meets specifications.

P(A) = 0.60   P(S|A) = 0.95
P(B) = 0.30   P(S|B) = 0.80
P(C) = 0.10   P(S|C) = 0.65

i) Probability that a regulator performs to specification (total probability)

P(S) = (0.60)(0.95) + (0.30)(0.80) + (0.10)(0.65)
     = 0.570 + 0.240 + 0.065
     = 0.875

ii) Probability that a regulator that meets specification came from B (Bayes' theorem)

P(B|S) = P(B) P(S|B) / P(S) = 0.240 / 0.875 = 0.2743

Answer: (i) 0.875, (ii) 0.2743.

✓ Answer checked by Claude Sonnet 5.5

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