Chapter 13.3, Problem 66AYU

To determine

To calculate: The probability that a household has an annual income of $\$50,000$ or more using following table which is based on a survey of annual incomes in $100$ households .

$\begin{array}{cccccc}\text{Income}& \$0-24,999& \$25,000-49,999& \$50,000-74,999& \$75,000-99,999& \$100,000\text{or}\text{more}\\ \begin{array}{l}\text{Number }\text{of }\\ \text{households}\end{array}& 22& 23& 17& 12& 26\end{array}$

Answer to Problem 66AYU

Solution:

The probability that a household has an annual income of $\$50,000$ or more is $0.55$.

Explanation of Solution

Given information:

The data of a survey of annual incomes in $100$ householdsis given in the following table.

$\boxed{\begin{array}{llllll}\text{Income}\hfill & \$0-24,999\hfill & \$25,000-49,999\hfill & \$50,000-74,999\hfill & \$75,000-99,999\hfill & \$100,000\text{or}\text{more}\hfill \\ \begin{array}{l}\text{Number}\text{of}\\ \text{households}\end{array}\hfill & 22\hfill & 23\hfill & 17\hfill & 12\hfill & 26\hfill \end{array}}$

Formula used:

If $E$ is any event and $S$ is the sample space then, the probability of event $E$ is, $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$.

Calculation:

The total number of households in a survey is $100$, so $n\left(S\right)=100$.

Let event $E$ denotes households with anannual income of $\$50,000$ or more.

According to given table, an income of $\$50,000$ or more represents three income groups;

$\$50,000-74,999$, $\$75,000-99,999$ and $\$100,000\text{or}\text{more}$.

There are $17$ households with an income group of $\$50,000-74,999$.

There are $12$ households with an income group of $\$75,000-99,999$.

There are $26$ households with an income group of $\$100,000$ or more.

Therefore the total number of households with an income of $\$50,000$ or more is $n\left(E\right)=17+12+26=55$.

Using the formula for probability of an event, $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$.

$\Rightarrow P\left(E\right)=\frac{55}{100}=0.55$

Therefore, the probability that a household has an annual income of $\$50,000$ or more is $0.55$.