Question

Get rich Refer to Exercise 63.

(a) Find P(“a good chance” | female).

(b) Find P(“a good chance”).

(c) Use your answers to (a) and (b) to determine whether the events “a good chance” and “female” are

independent. Explain your reasoning.

Short answer

Part (a) P (a good chance Female) =$0.2813$

Part (b) P ( a good chance) =$0.2944$

Part (c) Yes, independent.

Step-by-step solution

Part (a) Step 1. Given Information

In a recent year, the members of the United States Senate were described thus, according to the two-way table:

Consider the following events: D is a democrat, and F is a woman.

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 calculate the likelihood that the result for "a good chance" is correct.

Therefore,

$P(agoodchanceandFemale)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{663}{4826}$

$P\left(Female\right)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{2357}{4826}$

As a result, the conditional probability is:

$$\begin{gathered} P(agoodchanceFemale)=\frac{P(agoodchanceandFemale)}{P\left(Male\right)} \\ P(agoodchanceFemale)=\frac{663}{2357}\approx 0.2813 \end{gathered}$$

As a result, the likelihood that the result for "a good chance" is

$P(agoodchance/Female)=0.2813$

Part (b) Step 1. Calculation

The goal is to determine the probability that the result for "a good chance" is correct. Therefore, $$\begin{gathered} P(agoodchance)=\frac{favorableoutcomes}{possibleoutcomes} \\ P(agoodchance)=\frac{1421}{4826} \\ P(agoodchance)\approx 0.2944 \end{gathered}$$

Part (c) Step 1. Calculation 

part (a) and part (b) , $P(agoodchance/Female)=\frac{663}{2357}\approx 0.2813$

$P(agoodchance)=\frac{1421}{4826}\approx 0.2944$

They should have the same probability, hence they are independent.