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 by P(E₂/A).

Using Bayes’ theorem, we get
$$\begin{gathered} \mathrm{Required}\mathrm{probability}=P\left(E_2/A\right)=\frac{P\left(E_2\right)P\left(A/E_2\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)} \\ =\frac{0.4\times 0.3}{0.6\times 0.7+0.4\times 0.3} \\ =\frac{0.12}{0.54}=\frac{2}{9} \end{gathered}$$