Chapter 5.3, Problem 67E

a.

To determine

To find:the probability that the result for “democrats” in the given table.

Answer to Problem 67E

  $P\left(D|F\right)=0.7647$ .

Explanation of Solution

Given:

The two-way table describes the members of the U.S. Senate in a recent year.

Consider event D is democrat, even R is republic, F is female.

	Male	Female
Democrats	47	13
Republics	36	4

Concept used:

Probability is the number of favourable outcomes divided by the number of possible outcomes.

Thus, Definition condition probability:

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

Calculation:

According to the question, the person is female. Now to find the probability that the result for “democrats”.

Therefore,

  $P\left(\text{D and F}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{13}{47+13+36+4}=\frac{13}{100}$

  $P\left(\text{F}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{13+4}{100}=\frac{17}{100}$

Thus, the conditional probability becomes:

  $\begin{array}{l}P\left(D|F\right)=\frac{P\left(\text{D and F}\right)}{P\left(\text{F}\right)}\\ =\frac{13/100}{17/100}=\frac{13}{17}\approx 0.7647\end{array}$

Thus,the probability that the result for “democrats” is $P\left(D|F\right)=0.7647$ .

b.

To determine

To find: the probability that the result for “female” in the given table.

Answer to Problem 67E

  $P\left(F|D\right)=0.2167$ .

Explanation of Solution

Given:

The two-way table describes the members of the U.S. Senate in a recent year.

Consider event D is democrat, even R is republic, F is female.

	Male	Female
Democrats	47	13
Republics	36	4

Concept used:

Probability is the number of favourable outcomes divided by the number of possible outcomes.

Thus, Definition condition probability:

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

Calculation:

According to the question, the person is democrats. Now to find the probability that the result for “female”.

Therefore,

  $P\left(\text{D and F}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{13}{47+13+36+4}=\frac{13}{100}$

  $P\left(D\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{47+13}{100}=\frac{60}{100}$

Thus, the conditional probability becomes:

  $\begin{array}{l}P\left(F|D\right)=\frac{P\left(\text{D and F}\right)}{P\left(\text{D}\right)}\\ =\frac{13/100}{60/100}=\frac{13}{60}\approx 0.2167\end{array}$

Hence, the probability that the result for “female” $P\left(F|D\right)=0.2167$ .