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.
Let E1 and E2 be the respective events that the first group and the second group win the competition. 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
$\mathrm{P}\left(\mathrm{E}_{2} \mid \mathrm{A}\right)$
By using Bayes’ theorem, we obtain
$P\left(E_{2} \mid A\right)=\frac{P\left(E_{2}\right) \cdot P\left(A \mid E_{2}\right)}{P\left(E_{1}\right) \cdot P\left(A \mid E_{1}\right)+P\left(E_{2}\right) \cdot P\left(A \mid E_{2}\right)}$
$=\frac{0.4 \times 0.3}{0.6 \times 0.7+0.4 \times 0.3}$
$=\frac{0.12}{0.42+0.12}$
$=\frac{0.12}{0.54}$
$=\frac{12}{54}$
$=\frac{2}{9}$