Chapter 13.5, Problem 15E

To determine

To calculate: The probability that member chosen is a Democrat provided that member chosen is a man.

Answer to Problem 15E

The probability that member chosen is a Democrat provided that member chosen is a man is $\frac{1}{2}$ .

Explanation of Solution

Given information:

There are six democrats out of which two are women and six Republicans out of which 4 are men in a city council.

Formula used:

Probability of an event Eis $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Given an event Q , the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

Calculation:

Consider the provided information thatthere are six democrats out of which two are women and six Republicans out of which 4 are men in a city council.

There are 4 men who are Democrats and two women who are Republicans.

In total there are 8 men and 4 women. 12 members in city council.

The probability that member chosen is a Democrat provided that member chosen is a man.

Let A be an event that member chosen is a Democrat and B be an event that member chosen is a man.

Recall that probability of an event Eis $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Probability that member chosen is a man,

  $P\left(B\right)=\frac{8}{12}$

Probability that member chosen is a man and member chosen is a Democrat,

  $P\left(A\text{ and B}\right)=\frac{4}{12}$

Recall that given an event Q , the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

The probability that member chosen is a Democrat provided that member chosen is a man.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{4}{12}}{\frac{8}{12}}\\ =\frac{4}{8}\\ =\frac{1}{2}\end{array}$

Thus, the probability that member chosen is a Democrat provided that member chosen is a mans is $\frac{1}{2}$ .