Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired, whereas the other 40% must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?

Answer : Step 1 : Given twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired, whereas the other 40% must be replaced with new units. Let X be the number of phones submitted for service and Let R be be the number of telephones which are replaced after being submitted for service. We have the following distributions: X ~ Binomial (n = 10,p = 0.20)and X ~ Binomial (n = X,p = 0.40) x x x r Use the law of total probability to find the answer. Note that the first two probabilities in the sum are both zero, i.e., it is not possible to have. R=2 if x=0 or x=1. P(R = 2) = P R=2 P(X = 0) + P R=2 P(X = 1) + ... + P R=2 P(X = 10) ( X=0) (X=1 ) ( X=10) 10 R=2 P(R = 2) = P (X=x )P(X = x) x=2 10 x 2 102 10 x 10x P(R = 2) = ( )(p2) (1r p) ( x(p )x(1 p )x x=2 P(R = 2) = 0.1478070 Therefore the probability is 0.1478. Presumably, only phones that were submitted...