Question:
Two groups are competing for the position on the board 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 correspondind probability if 0.3,if the second group wins.Find the probability that the new product introduced was by the second group.
Two groups are competing for the position on the board 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 correspondind probability if 0.3,if the second group wins.Find the probability that the new product introduced was by the second group.
Updated On: Sep 21, 2023
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Solution and Explanation
The correct answer is: \(\frac{2}{9}\)
Given;
Probability that the first group wins the competition=\(P(G_1)=0.6\)
Probability that the second group wins the competition=\(P(G_2)=0.4\)
Let \(P\) denotes the lauching of new product.
\(∴(P|G_1)=0.7,(P|G_2)=0.3\)
By using Bayes’ theorem, we obtain;
\(P(G_1|P)=\frac{P(G_2)P(P|G_2)}{P(G_1)P(P|G_1)+P(G_2)P(P|G_2)}\)
\(=\frac{0.4×0.3}{0.6×0.7+0.4×0.3}\)
\(=\frac{12}{54}\)
\(=\frac{2}{9}\)
Given;
Probability that the first group wins the competition=\(P(G_1)=0.6\)
Probability that the second group wins the competition=\(P(G_2)=0.4\)
Let \(P\) denotes the lauching of new product.
\(∴(P|G_1)=0.7,(P|G_2)=0.3\)
By using Bayes’ theorem, we obtain;
\(P(G_1|P)=\frac{P(G_2)P(P|G_2)}{P(G_1)P(P|G_1)+P(G_2)P(P|G_2)}\)
\(=\frac{0.4×0.3}{0.6×0.7+0.4×0.3}\)
\(=\frac{12}{54}\)
\(=\frac{2}{9}\)
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Concepts Used:
Bayes Theorem
Bayes’ Theorem is a part of the conditional probability that helps in finding the probability of an event, based on previous knowledge of conditions that might be related to that event.
Mathematically, Bayes’ Theorem is stated as:-
\(P(A|B)=\frac{P(B|A)P(A)}{P(B)}\)
where,
- Events A and B are mutually exhaustive events.
- P(A) and P(B) are the probabilities of events A and B, respectively.
- P(A|B) is the conditional probability of the happening of event A, given that event B has happened.
- P(B|A) is the conditional probability of the happening of event B, given that event A has already happened.
This formula confines well as long as there are only two events. However, Bayes’ Theorem is not confined to two events. Hence, for more events.