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Q. 9 Two groups are competing for the position on 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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A: the first groups will win

B: the second groups will win

$P(A)=0.6$

$P(B)=0.4$

X: Event of introducing a new product.

Probability of introducing a new product if the first group wins : $P(X|A)=0.7$

Probability of introducing a new product if the second group wins : $P(X|B)=0.3$

$P(B|X)= \frac{P(B).P(X|B)}{P(B).P(X|B)+P(A).P(X|A)}$

$p(B|X) = \frac{0.4\times 0.3}{0.4\times 0.3+0.6\times 0.7}$

$p(B|X) = \frac{0.12}{0.12+0.42}$

$p(B|X) = \frac{0.12}{0.54}$

$p(B|X) = \frac{12}{54}$

$p(B|X) = \frac{2}{9}$

Hence, the probability that the new product introduced was by the second group :

$p(B|X) = \frac{2}{9}$

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seema garhwal

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