A carnival game consists of rolling a single fair die, with the following results: you win $8 for a 6, $7 for a 5, and lose $3 for any other number. Set a probability distribution, and calculate (already did) the
Expected value = .50
Variance= 24.58
standard deviation= 4.96
What fee should be charged to make it a fair game?
The equally probable monetary results of a toss are +8,+7,-3, -3, -3 and -3. The expected result is (15-12)/6 = 0.50, a 50 cent gain. I assume you calculated the variance correctly also.
The breakeven charge for a toss should be 50 cents. This is true regardless of the standard deviation. If the House charges more than 50 cents, the player will lose in the long run. Sixty cents would be a fair charge that allows a House profit. As a sucker bet, they might try to charge $1.
If all the possible outcomes are independent, one merely has to add the expected return of each outcome, so...
"fair game" charge = 8(1/6) + 7(1/6) + (-3)(4/6)
= .50
So in the "long run" at a 50 cents charge both the player and the house would break even.
To determine the fee that should be charged to make the game fair, we need to consider the expected value. The expected value represents the average amount of money you can expect to win or lose in a single game.
1. First, let's review the given probability distribution:
- Rolling a 6: You win $8 with a probability of 1/6.
- Rolling a 5: You win $7 with a probability of 1/6.
- Any other number: You lose $3 with a probability of 4/6.
2. Calculate the expected value:
To calculate the expected value, we multiply each outcome by its corresponding probability and sum them up:
Expected value = (8 * 1/6) + (7 * 1/6) + (-3 * 4/6) = 1.33 + 1.17 - 2 = $0.50
3. Charge a fee to make the game fair:
Since the expected value is $0.50, we need to adjust the game so that the expected value is $0.00 for it to be fair. In other words, players should, on average, neither win nor lose money in the long run.
To make the expected value $0.00, we can subtract the expected value from the amount won for rolling a 6:
$8 - $0.50 = $7.50
Therefore, a fee of $7.50 should be charged to make the game fair.