Question

Justin buys a racehorse for $25,000 and enters it in two races. He plans to sell the horse afterwards, hoping to make a profit. If the horse wins both races, its value will jump to $100,000. If the horse wins one of the races, it will be worth $60,000. If it loses both races, it will be worth only $15,000. Justin believes that there is a 25% chance the horse will win the first race and a 35% chance it will win the second race. Assume the two races are independent of one another. What is the the expected value?

Answer 1

1) Justin wins the first race, but loses the second race = (0.25) (1-0.35) = 0.1625 
2) Justin loses the first race, but wins the second race = (1-0.25) (0.35) = 0.2625 
3) Justin wins both races = (0.25) (0.35) = 0.0875 
4) Justin loses both races = (1-0.25) (1-0.35) = 0.4875 

If you add up all four probabilities, you will note that it equals 1.0. This means we did the calculations correctly. 

Expected value = ($60,000) (0.1625) + ($60,000) (0.2625) + ($100,000) (0.0875) + ($15,000) (0.4875) 

Expected value = $41,562.50 (these are his expected winnings from the races) 

Expected Profit = $41,562.50 - $25,000 = $16,562.50 

Answer 2

We will consider the following possibilities:
Wins none, wins first only, wins second only, wins both
a) Wins none:
Chances are: 75% x 65% = 48.75%
Thus, there is a 48.75% chance that the horse will sell at $15,000

b) Wins first only:
Chances are: 25% x 65% = 16.25%

c) Wins second only:
Chances are: 75% x 35% = 26.25%
Thus, the chances of the horse selling at $60,000 are 16.25 + 26.25
= 42.5%

d) Wins both:
Chances are: 25% x 35% = 8.75%
The chances that the horse will sell at $100,000 are 8.75%