Two group are competing for the position on the Board

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 Question:

                     Two group are competing for the position on the Board of Directors of a corporation. The probabilities that the first and second group will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probabilities 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 was introduced by the second group.

Solution:

Let E = The first group will win

F = The second group will win

A = The new product introduced

P(E) = 0.6, P(F) = 0.4

P(A/E) =  0.7, P(A/F) = 0.3

The probability that the new product was introduced by the second group

P(F/A) = \dfrac{P(F)\times \frac{A}{F}}{P(E)\times \frac{A}{E}+P(F)\times \frac{A}{F}}

= \dfrac{0.4 \times 0.3}{0.6 \times 0.7+0.4 \times 0.3}

= \dfrac{0.12}{0.42+0.12}

= \dfrac{0.12}{0.54}

= \dfrac{2}{9}

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