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, ifthe first group wins, the probability ofintroducing 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 ${E_1}:$ 'First group wins' and

${E_2}:$ 'Second group wins'

$\Rightarrow$ $P\left( {{E_1}} \right) = 0.6 = \cfrac{6}{{10}}$

and $P\left( {{E_2}} \right) = 0.4 = \cfrac{4}{{10}}$

Let $A$ : New product is introduced'

Then, $P\left( {A|{E_1}} \right) = 0.7 = \cfrac{7}{{10}}$ and $P\left( {A|{E_2}} \right) = 0.3 = \cfrac{3}{{10}}$

Hence, the required probability is

$P\left( {{E_2}|A} \right) = \cfrac{{P\left( {A|{E_2}} \right)P\left( {{E_2}} \right)}}{{P\left( {A|{E_1}} \right)P\left( {{E_1}} \right) + P\left( {A|{E_2}} \right)P\left( {{E_2}} \right)}}$

$= \cfrac{{\cfrac{3}{{10}} \times \cfrac{4}{{10}}}}{{\cfrac{7}{{10}} \times \cfrac{6}{{10}} + \cfrac{3}{{10}} \times \cfrac{4}{{10}}}} = \cfrac{{12}}{{42 + 12}} = \cfrac{{12}}{{54}} = \cfrac{2}{9}$