Chapter 10.4, Problem 5E

Solution

To find: The expected value of the given random variable.

The expected value is 2.625.

Given:

A spinner has 5-regions numbered 1 to 5.

The regions 1, 2, and 3 are each a quarter-circle.

The regions 4 and 5 divide the remaining area equally.

Concept used:

The expected value of a random variable is the sum of the products of the values of random variable with their corresponding probabilities.

Calculation:

Let $X$ be the random variable that gives region after spinning the spinner.

Since a quarter-circle is equal to $\frac{1}{4}$ , therefore

  $P\left(X=1\right)=P\left(X=2\right)=P\left(X=3\right)=\frac{1}{4}$ .

Now, the remaining area is

  $\begin{array}{c}1-\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right)=1-\frac{3}{4}\\ =\frac{1}{4}\end{array}$

Since the regions 4 and 5 divide the remaining area equally, therefore

  $\begin{array}{l}P\left(X=4\right)=P\left(X=5\right)=\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{2}\\ =\frac{1}{8}\end{array}$

Now, the expected value of the given random variable is,

  $\begin{array}{c}E\left(X\right)=1\left(\frac{1}{4}\right)+2\left(\frac{1}{4}\right)+3\left(\frac{1}{4}\right)+4\left(\frac{1}{8}\right)+5\left(\frac{1}{8}\right)\\ =\frac{1}{4}+\frac{1}{2}+\frac{3}{4}+\frac{1}{2}+\frac{5}{8}\\ =2.625\end{array}$

Hence, the expected value is 2.625.