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.

Given: let E_{1} be the event that first group wins the competition, E_{2} be the event that that second group wins the competition and A be the event of introducing a new product.

Then P(E_{1}) = 0.6

and P(E_{2}) = 0.4

Also P(A|E_{1}) = P (introducing a new product given that first group wins) = 0.7

And P(A|E_{2}) = P (introducing a new product given that second group wins) = 0.3

Now the probability of that new product introduced was by the second group, being given that a new product was introduced, is P(E_{2}|A).

By using bayes’ theorem, we have:

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