Chapter 20, Problem 1CS

To determine

To find: The probability model for the winnings and the expected winnings when the bet is made on red ball.

Answer to Problem 1CS

Solution: The probability model is described as follows:

$\begin{array}{ccc}\text{Bet Result}& \text{Win}& \text{Loose}\\ \text{Probability}& 0.4737& 0.5263\end{array}$

The expected value of bet is 5.3 cents.

Explanation of Solution

Calculation:

The bet is won if the ball lands in the red slot. There are a total of 18 red slots, so the probability of landing in a red slot is calculated as follows:

$\begin{array}{c}P\left(\text{Landing in red slot}\right)=\frac{18}{38}\\ =0.4737\end{array}$

The bet will be won if the ball lands in the red slot. So, the probability of winning will be same as that of the ball landing in the red slot. Hence, the probability of winning the bet is

$\begin{array}{c}P\left(\text{Winning the bet}\right)=P\left(\text{Landing in red slot}\right)\\ =\frac{18}{38}\\ =0.4737\end{array}$

The probability of losing the bet is calculated as follows:

$\begin{array}{c}P\left(\text{Losing the bet}\right)=1-P\left(\text{Winning the bet}\right)\\ =1-0.4737\\ =0.5263\end{array}$

The probability model is thus described as

$\begin{array}{ccc}\text{Bet Result}& \text{Win}& \text{Loose}\\ \text{Probability}& 0.4737& 0.5263\end{array}$

It is provided that if the ball lands in red slot then $2 is won along with the $1 spent on the bet. Thus, the expected amount won is computed as follows:

$\begin{array}{c}E\left(\text{bet}\right)=\left(\begin{array}{c}\left[\text{Amount won}\times P\left(\text{Winning the bet}\right)\right]+\\ \left[\text{Amount won}\times P\left(\text{Losing the bet}\right)\right]\end{array}\right)\\ =\left(1\times 0.4737\right)+\left(-1\times 0.5263\right)\\ =-0.0526\end{array}$