question_answer

Two groups are competing for the position on the board of directors of a corporation. The probability that the first and 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 introduce was by the second group.       

Answer:

Let E1 be the event that the first group wins, \[P\,({{E}_{1}})=0.6,\] \[{{E}_{2}}\]be the event that the second group wins. \[P\,({{E}_{2}})=0.4\]and A be the event where a new product in introduced P (introducing a new product, if the first group wins) \[=P\,\,\left( \frac{A}{{{E}_{1}}} \right)=0.7\] P (introducing a new product if the second group wins) \[=P\,\,\left( \frac{A}{{{E}_{2}}} \right)=0.3\] Now, using Baye?s theorem, we get \[=P\,\,\left( \frac{{{E}_{2}}}{A} \right)=\frac{P\,\,({{E}_{2}})\,\,P\,\left( \frac{A}{{{E}_{2}}} \right)}{P\,\,({{E}_{1}})\,\,P\,\left( \frac{A}{{{E}_{1}}} \right)+P\,\,({{E}_{2}})\,\,P\,\left( \frac{A}{{{E}_{2}}} \right)}\] \[=\frac{0.3\times 0.4}{(0.3)\,\,(0.4)+(0.7)\,\,(0.6)}\] \[=\frac{0.012}{0.012+0.042}=\frac{0.012}{0.054}=\frac{2}{9}\]