Chapter 11.8, Problem 30E

a.

To determine

To draw a Venn diagram.

Answer to Problem 30E

The Venn diagram is drawn as:

  

Explanation of Solution

Given information:

The numbers 1 through 20 are written on separate slips of paper and placed in a hat. One of the slips is randomly drawn.

The Venn diagram shows the possible outcomes for three events:

The number drawn is less than $5$

The number drawn is prime

The number drawn is a multiple of $3$

Let Event A be the number drawn is less than $5$

Let Event B be the number drawn is prime

Let Event C be the number drawn is a multiple of $3$

The Venn diagram is drawn as:

  

b.

To determine

To determine the probability of the events.

Answer to Problem 30E

The probability of getting a number less than $5$ or a prime number or a multiple of $3$ is $\text{P}\left(\text{A or B or C}\right)=\frac{9}{10}$ .

Explanation of Solution

Given information:

The numbers $1$ through $20$ are written on separate slips of paper and placed in a hat. One of the slips is randomly drawn.

The Venn diagram shows the possible outcomes for three events:

The number drawn is less than $5$

The number drawn is prime

The number drawn is a multiple of $3$

Formula used:

The formula used is:

  $\text{P}\left(\text{A or B or C}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)+\text{P}\left(\text{C}\right)-\text{P}\left(\text{A and B and C}\right)$

Calculation:

Let Event A be the number drawn is less than $5$

Let Event B be the number drawn is prime

Let Event C be the number drawn is a multiple of $3$

The probability of getting a number less than $5$ or a prime number or a multiple of $3$ from the numbers $1$ through $20$ is:

  $\begin{array}{l}\text{Probability of event A, P}\left(\text{A}\right)=\frac{5}{20}\\ \text{Probability of event B, P}\left(\text{B}\right)=\frac{8}{20}\\ \text{Probability of Event C, P}\left(\text{C}\right)=\frac{6}{20}\\ \text{Probability of }\left(\text{A and B and C}\right)\text{, P}\left(\text{A and B and C}\right)=\frac{1}{20}\\ \text{P}\left(\text{A or B or C}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)+\text{P}\left(\text{C}\right)-\text{P}\left(\text{A and B and C}\right)\left[\text{Probability of overlapping events}\right]\\ \text{P}\left(\text{A or B or C}\right)=\frac{5}{20}+\frac{8}{20}\text{+}\frac{6}{20}-\frac{1}{20}\left[\text{Simplify}\right]\\ \text{P}\left(\text{A or B or C}\right)=\frac{18}{20}\text{=}\frac{9}{10}\end{array}$

Hence, the probability of getting a number less than $5$ or a prime number or a multiple of $3$ is $\text{P}\left(\text{A or B or C}\right)=\frac{9}{10}$ .

c.

To determine

To determine the general rule for finding the probability of three overlapping events.

Answer to Problem 30E

The general rule for finding the probability of three overlapping events are:

  $\text{P}\left(\text{A or B or C}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)+\text{P}\left(\text{C}\right)-\text{P}\left(\text{A and B and C}\right)$

Explanation of Solution

Given information:

Three overlapping events are given

  $\text{P}\left(\text{A}\right),\text{P}\left(\text{B}\right),\text{P}\left(\text{C}\right)$

The general rule for finding the probability of three overlapping events are:

  $\text{P}\left(\text{A or B or C}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)+\text{P}\left(\text{C}\right)-\text{P}\left(\text{A and B and C}\right)$