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CBSE Class 12 Math 2009 Solved Paper

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Question : 23 of 29
Marks: +1, -0
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 was introduced by the second group.
Solution:  
Let E1E_1 be the event of the first group winning and E2E_2 be the event of the second group winning and S be the event of introducing a new product.
P(E1E_1) = 0.6 P(E2E_2) = 0.4
P (S|E1E_1) = 0.7
P (S|E2E_2) = 0.3
Probability of a new product being introduced by the second group will be , P (E2E_2|S)
P (E2E_2|S) =
P(E2)⋅P(S∣E2)P(E1)⋅P(S∣E1)+P(E2)⋅P(S∣E2)\frac{P(E_2) \cdot P(S|E_2)}{P(E_1) \cdot P(S|E_1) + P(E_2) \cdot P(S|E_2)}
= 0.4×0.30.4×0.3+0.7×0.6\frac{0.4 \times 0.3}{0.4 \times 0.3 + 0.7 \times 0.6}
= 0.120.12+0.42\frac{0.12}{0.12+0.42}
P (E2E_2|S) = 1254\frac{12}{54} = 29\frac{2}{9}
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