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Question

Two groups are competing for the positions of the Board of Directors of a Corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

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Solution

Let E1 and E2 denote the events that the first group and the second group win the competition, respectively. Let A be the event of introducing a new product.

P(E1) = Probability that the first group wins the competition = 0.6

P(E2) = Probability that the second group wins the competition = 0.4

P(A/E1) = Probability of introducing a new product if the first group wins = 0.7

P(A/E2) = Probability of introducing a new product if the second group wins = 0.3

The probability that the new product is introduced by the second group is given by P(E2/A).

Using Bayes’ theorem, we get
Required probability = PE2/A=PE2PA/E2PE1PA/E1+ PE2PA/E2 =0.4×0.30.6×0.7+0.4×0.3 =0.120.54=29


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