A pair of dice is rolled once. suppose you lose $9 if the dice sum to 9 and win $11 if the dice sum to 4 or 6. how much should you win or lose if any other number turns up in order for the game to be fair?
To determine the amount you should win or lose for any other number to make the game fair, we need to find the probabilities of rolling each number.
When rolling two dice, there are a total of 36 possible outcomes (6 outcomes for the first die multiplied by 6 outcomes for the second die).
Let's calculate the probabilities for each desired outcome:
1. If the dice sum to 9:
- There are four possible outcomes to sum to 9: (3, 6), (4, 5), (5, 4), (6, 3).
- The probability of each of these outcomes is 4/36 = 1/9.
- Since you lose $9 for this outcome, you should lose $9 with a probability of 1/9.
2. If the dice sum to 4 or 6:
- There are three possible outcomes to sum to 4: (1, 3), (2, 2), (3, 1).
- There are five possible outcomes to sum to 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1).
- The total number of favorable outcomes is 3 + 5 = 8.
- The probability of each of these outcomes is 8/36 = 2/9.
- Since you win $11 for these outcomes, you should win $11 with a probability of 2/9.
Now, to make the game fair, the expected value of the game should be zero. The expected value is calculated by multiplying each possible outcome by its probability and summing them up. In this case, we have:
(-$9) * (1/9) + (+$11) * (2/9) + x * (p) = 0,
where x represents the amount you should win or lose for any other number, and p represents the probability of any other number turning up.
Let's solve the equation for x:
(-$9) * (1/9) + (+$11) * (2/9) + x * (p) = 0,
(-$9/9) + ($22/9) + x * (p) = 0,
($13/9) + x * (p) = 0,
x * (p) = -($13/9),
x = -($13/9p).
So, you should win or lose -$13/9p for any other number to make the game fair.
To determine how much you should win or lose for any other number to make the game fair, we need to calculate the probability of each possible outcome and then distribute the total amount of money lost or won accordingly.
Let's start by calculating the probabilities:
- There are a total of 36 possible outcomes when rolling two dice (each die has 6 possible outcomes).
- The combinations that sum to 9 are:
- (3, 6), (4, 5), (5, 4), and (6, 3) - there are 4 possible outcomes.
- The combinations that sum to 4 or 6 are:
- (1, 3), (2, 2), (3, 1), (1, 5), (2, 4), (4, 2), (5, 1) and (2, 4) - there are 8 possible outcomes.
Now, let's calculate the probabilities of these outcomes:
- Probability of summing to 9: 4/36 = 1/9 (~0.1111)
- Probability of summing to 4 or 6: 8/36 = 2/9 (~0.2222)
Since the game is fair, the total amount of money lost (-$9) should be distributed proportionally to the probabilities of each outcome. Similarly, the total amount of money won (+$11) should also be distributed proportionally to the probabilities of each outcome.
First, let's calculate the fair amount to win or lose for each outcome:
- Loss per outcome (dice sum is 9): (-$9) / (1/9) = -$81
- Win per outcome (dice sum is 4 or 6): (+$11) / (2/9) = +$24.75
Now, let's determine the amount you should win or lose for any other number to make the game fair. Since there are 36 possible outcomes in total, we subtract the probabilities of summing to 9, 4, or 6 from 1 (to account for all other outcomes).
- Probability of any other outcome: 1 - (1/9 + 2/9) = 6/9 = 2/3
To distribute the remaining money fairly across all other outcomes:
- Fair amount to win or lose for any other number: (2/3) * (-$9) = -$6
Therefore, in order for the game to be fair, you should win or lose $6 for any other number that turns up.
There are 6 combinations for the first dice, and 6 combinations for the second dice, so the total number of possibilities is 36
There are only 4 ways to get a sum of 9: (4,5), (5,4), (3, 6), (6, 3)
There are only 3 ways to get 4: (2, 2), (1,3), (3,1)
There are only 5 ways to get 6: (1,5), (5,1), (2,4), (4,2), (3,3)
So there are 8 ways to get 4 or 6
There are 36-8-4 = 24 other possible ways to get another number
For the game to be fair, the sum of all the probabilities multiplied by the gain/loss for each of them should be 0:
-$9 *(4/36) + $11*(8/36) + x*(24/36) = 0
solve for x