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.

Open in App

Solution

Let $E_{1}$ and $E_{2}$ 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.

$P (E_{1}) =$ Probability that the first group wins the competition $= 0.6$
$P (E_{2}) =$ Probability that the second group wins the competition $= 0.4$
$P (A | E_{1}) =$ Probability of introducing a new product if the first group wins $= 0.7$
$P (A | E_{2}) =$ 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 $P (E_{2} | A)$ .
By using Baye's theorem, we obtain

$P (E_{2} | A) = \frac{P (E_{2}) \cdot P (A | E_{2})}{P (E_{1}) \cdot P (A | E_{1}) + P (E_{2}) \cdot P (A | E_{2})}$

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

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

$= \frac{0.12}{0.54}$

$= \frac{12}{54}$

$= \frac{2}{9} = 0.22$

Suggest Corrections

Similar questions

Q. Three groups

A, B, C

are contesting 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 groups

B

and

C

are

0.6

and

0.5

respectively. The probability that the new product will be introduced is given by

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 and C are 0.6 and 0.5 respectively. The probability that the new product will be introduced, is

Q. 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

0.07 k

, then

k = ?

Join BYJU'S Learning Program