The probability that an electrical switch which is kept in d

The probability that an electrical switch, which is kept in dry conditions, fails during the guarantee period, is 1%. If the switch is humid, the probability that it will fail during the guarantee period is 8%. Assume that 90% of switches are kept in dry conditions, whereas remaining 10% are kept in humid conditions. What is the probability that the switch fails during the guarantee period? If the switch failed during the guarantee period, what is the probability that it was kept in humid conditions?

Solution

Given P(D) = 90%=0.9 therefore Probability of switches are kept in wet condition P(W)=1-0.=0.1

P(F|D) =0.01; P(F|W)=0.08

a) Probability that the switch fails during the guarantee period is P(D)*P(F|D) + P(W)*P(F|W)

=0.09*0.01 + 0.1*0.08 = 0.026

b) If the switch failed during the guarantee period, the probability that it was kept in humid condition is

P(W|F) = [P(W and F)] /P(F) = (0.1*0.08)/0.026=0.307692