A machine produces photo detectors in pairs Tests show that

A machine produces photo detectors in pairs. Tests show that the first photo detector is acceptable with probability 0.6. When the first photo detector is acceptable, the second photo detector is acceptable with probability 0.8. If the first photo detector is defective, the second photo detector is acceptable with probability 0.4.
a)What is the probability that exactly one photo detector of a pair is acceptable?
b)What is the probability that both photo detectors are defective?

Solution

Let

A = photo detector 1 is acceptable
B = photo detector 2 is acceptable

and the ones with \' be the probability of defectives.

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a)

P(exactly one) = P(A) P(B\'|A) + P(A\') P(B|A\')

= 0.6(1 - 0.8) + (1 - 0.6)(0.4)

= 0.28 [ANSWER]

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b)

P(both are defective) = P(A\') P(B\'|A\')

= (1 - 0.6)(1 - 0.4)

= 0.24 [ANSWER]