Two groups are competing for the position on the board of directors of a corporation.
Question:

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.

Solution:

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

$P\left(E_{2} \mid A\right)=\frac{P\left(E_{2}\right) \cdot P\left(A \mid E_{2}\right)}{P\left(E_{1}\right) \cdot P\left(A \mid E_{1}\right)+P\left(E_{2}\right) \cdot P\left(A \mid E_{2}\right)}$

$=\frac{0.4 \times 0.3}{0.6 \times 0.7+0.4 \times 0.3}$

$=\frac{0.12}{0.42+0.12}$

$=\frac{0.12}{0.54}$

$=\frac{12}{54}$

$=\frac{2}{9}$

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