Question

A committee is composed of six Democrats and five Republicans. Three of the Democrats are men, and three of the Republicans are men. If a man is chosen for chairman, what is the probability that he is a Republican?

Short answer

The probability that the chairman is a Republican given that he is a man is \(\frac{1}{2}\) or 50%.

Step-by-step solution

Calculating the total number of members

Find the total number of members in the committee. There are 6 Democrats and 5 Republicans, therefore a total of 11 members.

Calculate the number of men

Find the total number of men in the committee. There are 3 male Democrats and 3 male Republicans, therefore a total of 6 men.

Calculate P(B), the probability that the chairman is a man

To calculate P(B), divide the total number of men by the total number of committee members: P(B) = (Number of Men) / (Total Number of Members) P(B) = 6 / 11

Calculate the number of male Republicans

We are given that there are 3 male Republicans in the committee.

Calculate P(A∩B), the probability that the chairman is a male Republican

To calculate P(A∩B), divide the total number of male Republicans by the total number of committee members: P(A∩B) = (Number of Male Republicans) / (Total Number of Members) P(A∩B) = 3 / 11

Calculate P(A|B), the probability that the chairman is a Republican given he is a man

Using the conditional probability formula, P(A|B) = P(A∩B) / P(B): P(A|B) = (3 / 11) / (6 / 11) The 11's cancel out: P(A|B) = 3 / 6

Simplify the probability

Simplify the probability by canceling out the common factor: P(A|B) = 1 / 2 The probability that the chairman is a Republican given that he is a man is 1/2 or 50%.