Question

Two groups are competing for the positions of 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.

 

Answer 1

Let E₁ and E₂ denote the events that the first group and the second group win the competition, respectively. Let A 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(A/E₁) = Probability of introducing a new product if the first group wins = 0.7

P(A/E₂) = Probability of introducing a new product if the second group wins = 0.3

The probability that the new product is introduced by the second group is given byP(E₂/A).

Using Bayes’ theorem, we get

\[\text{ Required probability } = P\left( E_2 /A \right) = \frac{P\left( E_2 \right)P\left( A/ E_2 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)}\]

\[ = \frac{0 . 4 \times 0 . 3}{0 . 6 \times 0 . 7 + 0 . 4 \times 0 . 3}\]

\[ = \frac{0 . 12}{0 . 54} = \frac{2}{9}\]