Chapter 4, Problem 139E

(a)

To determine

To find: The average of the payoff distribution for the provided bets.

Answer to Problem 139E

Solution: The average of the payoff distribution for the provided bets is $1/3$.

Explanation of Solution

Calculation: When the point 4 is obtained, then the probability of winning the bet is $1/3$. The distribution where 1 is the point and 0 is any other roll then,

Value of X	1	0
Probability	$1/3$	$2/3$

The mean of the distribution can be calculated as:

$\begin{array}{c}{\mu }_i=\Sigma x_i{p}_i\\ =\left(1\times \frac{1}{3}\right)\times \left(0\times \frac{2}{3}\right)\\ =\frac{1}{3}\end{array}$

When the point 5 or 9 is obtained, then the probability of winning is $2/5$. The distribution where 1 is the point and 0 is any other roll then,

Value of X	1	0
Probability	$2/5$	$3/5$

The mean of the distribution can be calculated as:

$\begin{array}{c}{\mu }_i=\Sigma x_i{p}_i\\ =\left(1\times \frac{2}{5}\right)\times \left(0\times \frac{3}{5}\right)\\ =\frac{2}{5}\end{array}$

When the point 5 or 9 is obtained, then the probability of winning is $2/5$. The distribution where 1 is the point and 0 is any other roll then,

Value of X	1	0
Probability	$2/5$	$3/5$

The mean of the distribution can be calculated as:

$\begin{array}{c}{\mu }_i=\Sigma x_i{p}_i\\ =\left(1\times \frac{2}{5}\right)\times \left(0\times \frac{3}{5}\right)\\ =\frac{2}{5}\end{array}$

When the point 6 or 8 is obtained, then the probability of winning is $5/11$. The distribution where 1 is the point and 0 is any other roll then,

Value of X	1	0
Probability	$5/11$	$6/11$

The mean of the distribution can be calculated as:

$\begin{array}{c}{\mu }_i=\Sigma x_i{p}_i\\ =\left(1\times \frac{5}{11}\right)\times \left(0\times \frac{6}{11}\right)\\ =\frac{5}{11}\end{array}$

(b)

To determine

To test: Whether the bets are fair and the average of the distribution of the payoff is 0.

Answer to Problem 139E

Solution: Yes, the bets are fair and the average of the distribution of the payoff is 0.

Explanation of Solution

Calculation: When betting $10 on any point, if bet is won by the individual he would receive $20, if the bet is lost by him then he would lost $10. Thus, the difference between bet amount and the average of the distribution of the payoff can be calculated as:

$\begin{array}{c}\text{Difference}=\left(20\times \frac{1}{3}\right)+\left(-10\times \frac{2}{3}\right)\\ =0\end{array}$

The difference between bet amount and the average of the distribution of the payoff is obtained 0.

When the point5 or 9 is obtained, then the probability of winning is $2/5$. The distribution where 1 is the point and 0 is any other roll then,

Value of X	1	0
Probability	$2/5$	$3/5$

The mean of the distribution can be calculated as:

$\begin{array}{c}{\mu }_i=\Sigma x_i{p}_i\\ =\left(1\times \frac{2}{5}\right)\times \left(0\times \frac{3}{5}\right)\\ =\frac{2}{5}\end{array}$

When betting $10 on any point, if bet is won by the individual he would receive $15, if the bet is lost by him then he would lost $10. Thus, the difference between bet amount and the average of the distribution of the payoff can be calculated as:

$\begin{array}{c}\text{Difference}=\left(15\times \frac{2}{5}\right)+\left(-10\times \frac{3}{5}\right)\\ =0\end{array}$

The difference between bet amount and the average of the distribution of the payoff is obtained 0.

When the point 6 or 8 is obtained, then the probability of winning is $5/11$. The distribution where 1 is the point and 0 is any other roll then,

Value of X	1	0
Probability	$5/11$	$6/11$

The mean of the distribution can be calculated as:

$\begin{array}{c}{\mu }_i=\Sigma x_i{p}_i\\ =\left(1\times \frac{5}{11}\right)\times \left(0\times \frac{6}{11}\right)\\ =\frac{5}{11}\end{array}$

When betting $10 on any point, if bet is won by the individual he would receive $12, if the bet is lost by him then he would lost $10. Thus, the difference between bet amount and the average of the distribution of the payoff can be calculated as:

$\begin{array}{c}\text{Difference}=\left(12\times \frac{5}{11}\right)+\left(-10\times \frac{6}{11}\right)\\ =0\end{array}$

The difference between bet amount and the average of the distribution of the payoff is obtained 0.

Conclusion: The difference between bet amount and the average of the distribution of the payoff is obtained as zero, if the difference were not 0 then the bet would not be considered as fair bet.