Chapter 12.5, Problem 33PPS

a.

To determine

To draw a Venn diagram to illustrate $\text{P}\left(\text{A and B}\right)$ .

Answer to Problem 33PPS

The Venn diagram can be drawn by:

  

Explanation of Solution

Given information:

Illustrate $\text{P}\left(\text{A and B}\right)$

The Venn diagram can be drawn by:

  

b.

To determine

To write the formula for $\text{P}\left(\text{B|A}\right)$ .

Answer to Problem 33PPS

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

Explanation of Solution

Given information:

The given statement is:

“Write the formula for $\text{P}\left(\text{B|A}\right)$ given the Venn diagram.”

Divide $\text{P}\left(\text{A and B}\right)$ by $\text{P}\left(\text{A}\right)$ :

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

Hence, the formula is $\text{P}\left(\text{B|A}\right)=\frac{\text{P}\left(\text{A and B}\right)}{\text{P}\left(\text{A}\right)}$ .

c.

To determine

To find the probability.

Answer to Problem 33PPS

The probability $\text{P}\left(\text{red}\right)=\frac{2}{3}\text{ or 67\% and P}\left(\text{red, green}\right)=\frac{8}{33}\text{ or 24\%}$ .

Explanation of Solution

Given information:

The given statement is:

“A jar contains $\text{12}$ marbles of which $\text{8}$ marbles are red and $\text{4}$ are green. If marbles are chosen without replacement, find $\text{P}\left(\text{red}\right)$ and $\text{P}\left(\text{red, green}\right)$ .”

Calculation :

The probability can be calculated by:

  $\begin{array}{l}\text{P}\left(\text{red}\right)=\frac{8}{12}=\frac{2}{3}\text{ or 67\%}\\ \text{P}\left(\text{red, green}\right)=\frac{2}{3}\times \frac{4}{11}\text{=}\frac{8}{33}\text{ or 24\%}\end{array}$

Hence, the probability $\text{P}\left(\text{red}\right)=\frac{2}{3}\text{ or 67\% and P}\left(\text{red, green}\right)=\frac{8}{33}\text{ or 24\%}$ .

d.

To determine

To find the probability on choosing a green marble on the second selection.

Answer to Problem 33PPS

The probability of choosing a green marble on the second selection, given that the first marble selected was red is $\frac{4}{11}\text{ or 36\%}$ .

Explanation of Solution

Given information:

The given statement is:

“Using the probabilities from part c and the Venn diagram in part a, determine the probability of choosing a green marble on the second selection, given that the first marble selected was red.”

Calculation :

The probability can be calculated by:

  $\begin{array}{l}\text{P}\left(\text{green|red}\right)=\frac{\frac{8}{33}}{\frac{2}{3}}=\frac{8}{33}\times \frac{3}{2}=\frac{4}{11}\\ \text{P}\left(\text{green|red}\right)=\frac{4}{11}\text{ or 36\%}\end{array}$

Hence, the probability of choosing a green marble on the second selection, given that the first marble selected was red is $\frac{4}{11}\text{ or 36\%}$ .

e.

To determine

To write the formula forfinding conditional probability.

Answer to Problem 33PPS

The formula for finding conditional probability is $\text{P}\left(\text{B|A}\right)=\frac{\text{P}\left(\text{A and B}\right)}{\text{P}\left(\text{A}\right)}$ .

Explanation of Solution

Given information:

The given statement is:

“Write the formula for finding a conditional probability.”

The formula is:

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

Hence, the formula for finding conditional probability is $\text{P}\left(\text{B|A}\right)=\frac{\text{P}\left(\text{A and B}\right)}{\text{P}\left(\text{A}\right)}$ .

f.

To determine

To find the probability that a fan waved a banner and cheered for the home team.

Answer to Problem 33PPS

The probability that a fan waved a banner and cheered for the home team is $0.25\text{ or }25\%$ .

Explanation of Solution

Given information:

The given statement is:

“At a basketball game, $\text{80\%}$ of the fans cheered for the home team. In the same crowd, $\text{20\%}$ of the fans were waving banners and cheering for the home team.”

Formula used:

The following formula is used:

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

Calculation:

The probability can be calculated by:

  $\begin{array}{l}\text{P}\left(\text{B|A}\right)=\frac{\text{P}\left(\text{A and B}\right)}{\text{P}\left(\text{A}\right)}\\ \text{P}\left(\text{B|A}\right)=\frac{0.20}{0.80}\\ \text{P}\left(\text{B|A}\right)=0.25\text{ or }25\%\end{array}$

Hence, the probability that a fan waved a banner and cheered for the home team is $0.25\text{ or }25\%$ .