At this year's State Fair, there was a dice rolling game. If you rolled two dice and got a sum of 2 or 12, you won $20. If you rolled a 7,you won $5. Any other roll was a loss. It cost $3 to play one game with one roll of the dice. What is the expectation of the game?

Question

At this year's State Fair, there was a dice rolling game. If you rolled two dice and got a sum of 2 or 12, you won $20. If you rolled a 7,you won $5. Any other roll was a loss. It cost $3 to play one game with one roll of the dice. What is the expectation of the game?

Answer 1

Calculate the product of the net gain (i.e. gain minus the bet of $3) and the probability of each outcome.

Add all the products to get the expectation of the game.
Note that the probabilities of all the outcomes must add up to exactly 1.0

Answer 2

To calculate the expectation of the game, we need to find the average amount of money won or lost per game.

First, let's calculate the probabilities of getting each possible outcome when rolling two dice:

- There is only one way to get a sum of 2: by rolling both dice as 1 (1,1). The probability of this occurring is 1/36.
- There is only one way to get a sum of 12: by rolling both dice as 6 (6,6). The probability of this occurring is also 1/36.
- To get a sum of 7, we can roll the following combinations: (1,6), (2,5), (3,4), (4,3), (5,2), or (6,1). There are a total of 6 possible combinations to get a sum of 7. The probability of rolling a sum of 7 is thus 6/36, which reduces to 1/6.
- For any other outcome, the probability is 1 - (probability of getting a sum of 2 or 12) - (probability of getting a sum of 7). So, it is 1 - (1/36) - (1/6) = 29/36.

Now, let's calculate the expected value per game:

- If you roll a sum of 2 or 12, you win $20. The probability of rolling a sum of 2 or 12 is (1/36) + (1/36) = 2/36, which reduces to 1/18. So, the expected value of winning $20 is (1/18) * $20 = $1.11.
- If you roll a sum of 7, you win $5. The probability of rolling a sum of 7 is 1/6. So, the expected value of winning $5 is (1/6) * $5 = $0.83.
- For any other roll, you lose $3. The probability of any other outcome is 29/36. So, the expected value of losing $3 is (29/36) * (-$3) = -$2.36.

Now, let's calculate the overall expected value of the game by adding up these individual expected values:

Expected value = ($1.11) + ($0.83) + (-$2.36) = -$0.42

Therefore, the expectation of the game is -$0.42, which means that, on average, you would lose approximately 42 cents per game.