Chapter 11.3, Problem 4GP

To determine

To find: The expected value of the winnings.

Answer to Problem 4GP

The expected value is $\$5.57$ .

Explanation of Solution

Given:

The table is given below:

Value ($)	$1$	$10$	$100$	$1000$	$5000$	$25,000$
Frequency	$5000$	$100$	$25$	$5$	$1$	$1$

The theoretical probability is calculated by dividing the number of winners for each prize by $5132$ as:

Value $X$	$1$	$10$	$100$	$1000$	$5000$	$25,000$
Frequency	$5000$	$100$	$25$	$5$	$1$	$1$
$P\left(X\right)$	$0.97$	$0.02$	$0.004$	$0.001$	$0.0001$	$0.0001$

The expected value is given as:

  $E\left(X\right)={\displaystyle \sum \left[X\cdot P\left(X\right)\right]}$

Here, $X=\text{prizes, }P=\text{probability of corresponding prize}$ .

So the expected value is calculated as:

  $\begin{array}{l}E\left(X\right)=\$1\left(0.97\right)+\$10\left(0.02\right)+\$100\left(0.004\right)+\$1000\left(0.001\right)\\ +\$5000\left(0.0001\right)+\$25000\left(0.0001\right)\\ E\left(X\right)=\$5.57\end{array}$

Therefore the expected value is $\$5.57$ .