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Question
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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Solution
Let E1: 'First group wins' and E2: 'Second group wins'
⇒ `P (E_1) = 0.6 = 6/10`
and `P (E_2) = 0.4 = 4/10`
Let A: 'New product is introduced`
Then `P (A|E_1) = 0.7 = 7/10`
and `P (A|E_2) = 0.3 = 3/10`
Hence the required probability is
`= P (E_2 |A) = (P(A|E_2) P (E_2))/(P (A|E_1) P (E_1) + P (A| E_2) P (E_2))`
`= (3/10 xx 4/10)/(7/10 xx 6/10 + 3/10 xx 4/10)`
`= 12/(42 + 12)`
`= 12/54`
`= 2/9`
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