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 E₁ be the event of the first group winning and E₂ be the event of the second group winning and S 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 ( S $Ι$ E₁ ) = Probability of introducing a new product if the first group wins = 0.7

p ( S $Ι$ E₂ ) = Probability of introducing a new product if the second group wins = 0.3

The probability taht the new product is introduced by the second group is given by P ( E₂ $Ι$ S ).

By using Bayes' theorem, we obtain

P ( E₂ $Ι$ S = $\frac{P (E_{2}) . P (S Ι E_{2})}{P (E_{1}) . P (S Ι E_{1}) + P (E_{2}) . P (S Ι E_{2})}$

= $\frac{0 . 4 x 0 . 3}{0 . 4 x 0 . 3 + 0 . 7 x 0 . 6}$

$= \frac{0 . 12}{0 . 12 + 0 . 42} = \frac{0 . 12}{0 . 54} = \frac{12}{54} = \frac{2}{9}$

P ( E₂ $Ι$ S ) = $\frac{2}{9}$ .

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