Suppose you win $3 if you roll a sum of 2,3,4, or 5, you lose $15 if you roll a sum of 6,7, or 8, and you win $5 if you roll a sum of 9,10,11, or 12. Find the expected value (mean of the probability distribution) of how much you win or lose for the game.
I will assume you are rolling 2 dice ....
prob(2,3,4,or 5) 1/36 + 2/36 + 3/36 + 4/36 = 5/18
prob(6,7,or8) = 5/36 + 6/36 + 5/35 = 8/18
prob(9,10,11 or 12) = 4/36 + 3/36+ 2/36 + 1/36 = 5/18
expected value = (5/18)(3) - (8/18)(15) + (5/18)(5) = $ -4.44
Bad game, don't play it
Well, I must say, that's quite the gamble you've got there! Let's crunch the numbers and find out the expected value, shall we?
First, we need to calculate the probabilities of rolling each sum. There are 36 possible outcomes when rolling two six-sided dice, so let's break it down:
- Sum of 2: There is only one way to roll a 2 (1+1), so the probability is 1/36.
- Sum of 3: There are two ways to roll a 3 (1+2 or 2+1), so the probability is 2/36 = 1/18.
- Sum of 4: There are three ways to roll a 4 (1+3, 2+2, or 3+1), so the probability is 3/36 = 1/12.
- Sum of 5: There are four ways to roll a 5 (1+4, 2+3, 3+2, or 4+1), so the probability is 4/36 = 1/9.
- Sum of 6: There are five ways to roll a 6 (1+5, 2+4, 3+3, 4+2, or 5+1), so the probability is 5/36.
- Sum of 7: There are six ways to roll a 7 (1+6, 2+5, 3+4, 4+3, 5+2, or 6+1), so the probability is 6/36 = 1/6.
- Sum of 8: There are five ways to roll an 8 (2+6, 3+5, 4+4, 5+3, or 6+2), so the probability is 5/36.
- Sum of 9: There are four ways to roll a 9 (3+6, 4+5, 5+4, or 6+3), so the probability is 4/36 = 1/9.
- Sum of 10: There are three ways to roll a 10 (4+6, 5+5, or 6+4), so the probability is 3/36 = 1/12.
- Sum of 11: There are two ways to roll an 11 (5+6 or 6+5), so the probability is 2/36 = 1/18.
- Sum of 12: There is only one way to roll a 12 (6+6), so the probability is 1/36.
Now, let's calculate the expected value using the probabilities and payouts:
Expected value = (1/36 * 3) + (1/18 * 3) + (1/12 * 3) + (1/9 * 3) + (5/36 * -15) + (1/6 * -15) + (5/36 * -15) + (4/36 * 5) + (1/9 * 5) + (1/12 * 5) + (1/18 * 5) + (1/36 * 5)
Calculating this gives us an expected value of approximately -$1.25.
So, on average, you can expect to lose about $1.25 per game. But hey, at least you'll always have a good time rolling the dice!
To find the expected value, we need to calculate the probability of each possible outcome and multiply it by the corresponding payout.
Let's find the probabilities for each possible sum:
- The sum of 2 can only be rolled with a combination of (1,1), which has a probability of 1/36.
- The sum of 3 can be rolled with (1,2) or (2,1), which each have a probability of 1/36.
- The sum of 4 can be rolled with (1,3), (2,2), or (3,1), which each have a probability of 1/36.
- The sum of 5 can be rolled with (1,4), (2,3), (3,2), or (4,1), which each have a probability of 1/36.
- The sum of 6 can be rolled with (1,5), (2,4), (3,3), (4,2), (5,1), which each have a probability of 1/36.
- The sum of 7 can be rolled with (1,6), (2,5), (3,4), (4,3), (5,2), or (6,1), which each have a probability of 1/36.
- The sum of 8 can be rolled with (2,6), (3,5), (4,4), (5,3), or (6,2), which each have a probability of 1/36.
- The sum of 9 can be rolled with (3,6), (4,5), (5,4), or (6,3), which each have a probability of 1/36.
- The sum of 10 can be rolled with (4,6), (5,5), or (6,4), which each have a probability of 1/36.
- The sum of 11 can be rolled with (5,6) or (6,5), which each have a probability of 1/36.
- The sum of 12 can only be rolled with (6,6), which has a probability of 1/36.
Now let's calculate the expected value:
Expected value = (P(2) * $3) + (P(3) * $3) + (P(4) * $3) + (P(5) * $3) + (P(6) * (-$15)) + (P(7) * (-$15)) + (P(8) * (-$15)) + (P(9) * $5) + (P(10) * $5) + (P(11) * $5) + (P(12) * $5)
Expected value = (1/36 * $3) + (2/36 * $3) + (3/36 * $3) + (4/36 * $3) + (5/36 * (-$15)) + (6/36 * (-$15)) + (5/36 * (-$15)) + (4/36 * $5) + (3/36 * $5) + (2/36 * $5) + (1/36 * $5)
Calculating the values:
Expected value = ($3/36) + ($6/36) + ($9/36) + ($12/36) - ($75/36) - ($90/36) - ($75/36) + ($20/36) + ($15/36) + ($10/36) + ($5/36)
Simplifying the fractions:
Expected value = ($3 + $6 + $9 + $12 - $75 - $90 - $75 + $20 + $15 + $10 + $5) / 36
Calculating the sum:
Expected value = (-$180 + $72) / 36
Expected value = (-$108) / 36
Expected value = -$3
Therefore, the expected value for this game is -$3. This means that, on average, you are expected to lose $3 per game.
To find the expected value of how much you win or lose for the game, we need to calculate the probability of each possible outcome and multiply it by the corresponding amount of money won or lost.
Let's calculate the probabilities for each sum:
- Sum of 2, 3, 4, or 5: To get a sum of 2, there is only one possible outcome (1,1). For a sum of 3, there are two possible outcomes (1,2) and (2,1). Similarly, there are three possible outcomes each for sums of 4 and 5, which are (1,3), (2,2), (3,1) and (1,4), (2,3), (3,2), (4,1) respectively. In total, there are 10 possible outcomes for sums of 2, 3, 4, or 5. The probability of each sum is 10/36 = 5/18.
- Sum of 6, 7, or 8: To get a sum of 6, there are five possible outcomes (1,5), (5,1), (2,4), (4,2), and (3,3). Similarly, there are six possible outcomes each for sums of 7 and 8, which are (1,6), (6,1), (2,5), (5,2), (3,4), (4,3) and (2,6), (6,2), (3,5), (5,3), (4,4), (6,2) respectively. In total, there are 17 possible outcomes for sums of 6, 7, or 8. The probability of each sum is 17/36.
- Sum of 9, 10, 11, or 12: To get a sum of 9, there are four possible outcomes (3,6), (6,3), (4,5), and (5,4). Similarly, there are three possible outcomes each for sums of 10 and 11, which are (4,6), (6,4), (5,5) and (5,6), (6,5), (6,6) respectively. In total, there are 10 possible outcomes for sums of 9, 10, 11, or 12. The probability of each sum is 10/36 = 5/18.
Now, let's calculate the expected value:
Expected value = (Probability of winning $3) * ($3) + (Probability of losing $15) * (-$15) + (Probability of winning $5) * ($5)
Expected value = [(5/18) * $3] + [(17/36) * (-$15)] + [(5/18) * $5]
Expected value = $5/6 - $17/4 + $25/18
Expected value = ($60 - $153 + $25)/36
Expected value = -$68/36
Expected value = -$1.89
Therefore, the expected value of how much you win or lose for the game is -$1.89. This means that, on average, you can expect to lose $1.89 each time you play this game.