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Chapter 2: Problem 111

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

Expert verified
The probability that a man chosen as chairman will be a Republican is \(\frac{1}{2}\) or 50%.

Step by step solution

01

Total number of people in the committee

The committee is composed of 6 Democrats and 5 Republicans, making a total of 11 people in the committee.
02

Number of men in the committee

Three of the Democrats are men, and three of the Republicans are also men. So, there are a total of 3+3 =6 men in the committee.
03

Probability of choosing a man as chairman

Since there are 6 men in the committee of 11 people, the probability of choosing a man as chairman is given by the ratio of men to the total number of people: \(P(Man) = \frac{6}{11}\)
04

Finding the probability of choosing a Republican man, given that a man is the chairman

Now we are given the condition that a man is chosen as the chairman. There are 3 Republican men in the committee, so the probability of choosing a Republican man, given that a man is the chairman, is given by the ratio of Republican men to the total number of men: \(P(Republican\ Man | Man) = \frac{3}{6}\)
05

Calculating the conditional probability

We can now use the conditional probability formula to find the probability of choosing a Republican man as chairman, given that a man is the chairman: \(P(Republican\ Man | Man)= \frac{P(Republican\ Man\cap Man)}{P(Man)}\) Since \(P(Republican\ Man \cap Man)\) is the same as \(P(Republican\ Man)\), we can rewrite the equation: \(P(Republican\ Man)= \frac{P(Republican\ Man | Man)\times P(Man)}\) Now, plug in the probabilities we found in Step 3 and Step 4: \(P(Republican\ Man) = \frac{(\frac{3}{6})\times (\frac{6}{11})}{(\frac{6}{11})}\)
06

Simplifying and finding the final probability

We can now simplify the equation and find the final probability: \(P(Republican\ Man) = \frac{3}{6} = \frac{1}{2}\) So, there is a 50% chance that a man chosen as chairman will be a Republican.

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