Question

Sampling senators The two-way table describes the members of the U.S. Senate in a recent year. Suppose we select a senator at random. Consider events D: is a democrat, and F: is female.

(a) Find P(D | F). Explain what this value means.

(b) Find P(F | D). Explain what this value means.

Short answer

Part (a)P (D|F) $=0.7647$

Part (b) $P\left(F\right|D)=0.2167$

Step-by-step solution

Part (a) Step 1. Given Information

The results are distributed as follows:

Part (a) Step 2. Concept Used

The number of favorable outcomes divided by the total number of possible outcomes equals probability. As a result, the following is a definition of condition probability: $P\left(A\right|B)=\frac{P(AandB)}{P\left(B\right)}$

Part (a) Step 3. Calculation

The person in issue is female, according to the inquiry. Now we must determine the likelihood that the result for "democrats" will be positive.

Therefore,

$P(DandF)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{13}{47+13+36+4}=\frac{13}{100}$

$P\left(F\right)=\frac{favorableoutcomes}{possibleoutcome}\frac{13+4}{100}=\frac{17}{100}$

As a result, the conditional probability is:

$P\left(D\right|F)=\frac{P(DandF)}{P\left(F\right)}=\frac{13}{17}\approx 0.7647$

As a result, the likelihood of a favorable outcome for "democrats" is high.

$P\left(D\right|F)=0.7647$

Part (b) Step 1. Calculation

The person is a democrat, according to the question. Now we must determine the likelihood that the result for "female" is correct.

Therefore,

$P(DandF)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{13}{47+13+36+4}=\frac{13}{100}$

$P\left(D\right)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{47+13}{100}=\frac{60}{100}$

As a result, the conditional probability is:

$P\left(F\right|D)=\frac{P(DandF)}{P\left(D\right)}=\frac{13}{60}\approx 0.2167$

As a result, the likelihood that the "female" result will be positive is high.

$P\left(F\right|D)=0.2167$