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प्रश्न
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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उत्तर
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 byP(E2/A).
Using Bayes’ theorem, we get
\[\text{ Required probability } = P\left( E_2 /A \right) = \frac{P\left( E_2 \right)P\left( A/ E_2 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)}\]
\[ = \frac{0 . 4 \times 0 . 3}{0 . 6 \times 0 . 7 + 0 . 4 \times 0 . 3}\]
\[ = \frac{0 . 12}{0 . 54} = \frac{2}{9}\]
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