Chapter 16, Problem 8E

To determine

To find: the expected profit of men.

Answer to Problem 8E

$10600

Explanation of Solution

Given:

  $\begin{array}{c}P\left(F\right)=0.2\\ P\left(S\right)=0.3\\ P\left(F\text{'}\right)=1-0.2\\ =0.8\\ P\left(S\text{'}\right)=1-0.3\\ =0.7\end{array}$

Formula used:

  $E\left(X\right)={\displaystyle \sum xP\left(X=x\right)}$

Calculation:

F= Horse wins first race

S = Horse wins second race

Probability of horse wins both the races is

  $\begin{array}{c}P\left(F\cap S\right)=P\left(F\right)P\left(S\right)\\ =\left(0.2\right)\left(0.3\right)\\ =0.06\end{array}$

Probability of horse wins one of the two races is

  $\begin{array}{c}P\left(F\cap S\text{'}\right)=P\left(F\text{'}\cap S\right)\\ =\left(0.2\right)\left(0.7\right)+\left(0.8\right)\left(0.3\right)\\ =0.14+0.24\\ =0.38\end{array}$

Probability of horses lose both the races is

  $\begin{array}{c}P\left(F\text{'}\cap S\text{'}\right)=\left(0.8\right)\left(0.7\right)\\ =0.56\end{array}$

Following is probability model

X	$100,000	$50,000	10,000
P(X = x)	0.06	0.38	0.56

There is subtracted $20,000 from expected value although it is money spent to purchase the horse.

Expected profit is

  $\begin{array}{c}E\left(X\right)={\displaystyle \sum xp\left(x\right)-20,000}\\ =\$100000\left(0.06\right)+\$500000\left(0.56\right)-\$20000\\ =\$6000+\$19000+\$5600-\$20000\\ =\$10,600\end{array}$

Expected profit is $10,600