For a math assignment we need to design a dice roll game with different expected values of winning. The game should require a player to make a bet on the sum of one roll of two dice. If the dice sum to that number, the player wins; otherwise, he loses. We need to create a fair game where the expectation of winning is zero, and a game in the house's favor, where the expectation of winning is slightly negative.
So far I found the probability distribution of X, where X is the sum of the dice:
2: P(X=x)= .0278
3: P(X=x)= .0556
4: P(X=x)= .0833
5: P(X=x)= .1111
6: P(X=x)= .1389
7: P(X=x)= .1667
8: P(X=x)= .1389
9: P(X=x)= .1111
10: P(X=x)= .0833
11: P(X=x)= .0556
12: P(X=x)= .0278
I also found the expected value of X: 7.
So from this I know that probability of each sum differs, with rolling a sum of 7 being more probable than rolling a sum of 2 or 12.
I'm having some trouble figuring out the expected value of the game, however. Say that to play my game, a person has to pay $1. If he wins, he wins $6, making the net outcome of a win $5. So the two outcomes are: -1 and 5. The way we found the expected value in class was we found the outcomes and the probability of each outcome, multiplied the outcome by the probability, and summed the products. The part that is confusing me is, doesn't the probability of winning depend on the sum that a player guesses? Isn't someone who guesses 7 as their sum going to be more likely to win than someone who guesses a 2?
Am I overthinking this or going about this in the wrong way? Can someone help me figure this out?
To calculate the expected value of the game, you need to consider the probability of winning for each sum that a player can bet on. You're correct in understanding that the probability of winning does depend on the sum that a player guesses.
To calculate the expected value of the fair game, where the expectation of winning is zero, you need to find the amount that a player wins or loses for each possible sum and multiply it by the probability of that sum occurring. In this case, since the player pays $1 to play and wins $6 if they guess correctly, the outcomes are -1 (player loses $1) and 5 (player wins $6, net outcome of $5).
Now, let's calculate the expected value step by step:
1. Multiply the outcome (-1 or 5) by the corresponding probability for each sum. For example, for the sum of 2, the outcome is -1, and the probability is 0.0278. Multiply -1 by 0.0278.
2. Repeat this calculation for each sum and add up the results. This will give you the sum of the products.
3. Finally, divide the sum of the products by the number of possible sums (in this case, there are 11 possible sums) to obtain the expected value.
By following these steps, you'll calculate the expected value of the fair game, which should be zero since it is a fair game.
To create a game where the expectation of winning is slightly negative, you can adjust the payoff amounts for winning and losing. For example, you could increase the amount the player pays to play the game or decrease the amount they win if they guess correctly. By modifying these values, you can create a game where the expected value is negative.
Remember that the expected value represents the average outcome over a large number of plays. It does not guarantee that a player will win or lose a specific amount in a single play, but rather provides an estimate of the long-term average outcome.
I hope this explanation helps you understand how to calculate the expected value of the game and how to create variations with different expected values. Let me know if you have any further questions!