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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.
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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