Question

Three groups a, B & C are competing for positions on the Board of directors of a company. The probabilities of their winning are 0.5, 0.3, 0.2 respectively. If the group A wins, the probability of introducing a new product is 0.7 and the corresponding probabilities for group B & C are 0.6 & 0.5 respectively. Find the probability that the new product will be introduced.
If ans can be expressed as $0.07k$, then $k=?$

Answer 1

Let $E$ be the event that a new product is inthroduced
Then $P\left(A\right)=0.5,P\left(B\right)=0.3,P\left(C\right)=0.2$
and $P\left(\frac{E}{A}\right)=0.7,P\left(\frac{E}{B}\right)=0.6,P\left(\frac{E}{C}\right)=0.5$
Therefore $A,B$ and $C$ are mutually exclusive and exhaustive events
$P\left(E\right)=P\left(A\right)\times P\left(\frac{E}{A}\right)+P\left(B\right)\times P\left(\frac{E}{A}\right)+P\left(C\right)\times P\left(\frac{E}{C}\right)=0.5\times 0.7+0.3\times 0.6+0.2\times 0.5=0.35+0+.18+0.10=0.63$