Question

Have you ever betted on a horse race? Suppose you try and it costs you $25 to bet on a horse race. There are 8 horses in the race with equal chances of winning. You win $125 if your horse wins. If your horse places 2nd or 3rd you receive your money back. If your horse places 4th through 8th you lose your money. Find your expected net gain (or loss) for playing one game.

a. -$6.25 b. $0 c. $-0.25 d. -$3.25

Answer 1

To determine the expected net gain or loss for a horse race bet, one must multiply the outcomes by their probabilities and sum them. In this case, the expected value is zero, meaning there is no expected gain or loss over the long term for this particular game.

Step-by-step explanation:

To calculate the expected net gain or loss from betting on a horse race with the given conditions, we need to consider the probabilities of each outcome and the gains or losses associated with them. If there are 8 horses, each has an equal chance of 1 in 8, or 0.125, of winning, placing 2nd, or placing 3rd.

Therefore, the probability of winning ($125 gain) is 0.125, the probability of either placing 2nd or 3rd (no gain or loss, $0) is 2 × 0.125 since there are two such outcomes, and the probability of placing 4th through 8th (losing the $25 bet) is 5 × 0.125 because there are five such outcomes.

Now we calculate the expected value (EV), which is found by multiplying each outcome's gain or loss by its probability and then summing these products:

• EV(win) = 0.125 × $125
• EV(place) = 0.125 × 2 × $0 (since it's just getting the bet back)
• EV(lose) = 0.125 × 5 × (-$25)

The total expected value is the sum of these individual expected values:

EV(total) = EV(win) + EV(place) + EV(lose)

EV(total) = (0.125 × $125) + (0.25 × $0) + (0.625 × -$25)

EV(total) = $15.625 + $0 - $15.625

EV(total) = $0

The expected net gain or loss for playing one game in this scenario is $0, therefore the correct answer is b. $0.

Answer 2

The expected net loss for betting on the horse race is approximately -$1.95 per game, calculated using the probabilities of the different outcomes and their respective payoffs.

Step-by-step explanation:

The student asked about the expected net gain or loss for placing a $25 bet on a horse race. To calculate this, we consider each outcome's probability and associated payoff.

Winning (1st place) with one out of eight horses gives a probability of 1/8 and a payoff of $125.

Placing 2nd or 3rd gives back the $25 bet, with a combined probability for these two places of 2/8 since there are two chances out of eight to not lose money.

Placing 4th through 8th results in a loss of the $25 bet, with a probability of 5/8 because there are five ways this can happen out of eight possible outcomes.

Let's calculate the expected value (EV):
EV = (1/8 * $125) + (2/8 * $0) - (5/8 * $25)
EV = ($15.625) + ($0) - ($15.625)
EV = $0 - $15.625
EV = -$15.625 / 8

Divide the total gain or loss by the number of possible outcomes (8):
EV = -$1.953125
Thus, the expected net gain or loss is approximately -$1.95, which is not one of the provided options. So there seems to be an error in the question. The correct calculation indicates that, for every game played, the player would expect to lose about $1.95 on average.