```We purpose to construct a dice game that involves rolling one die in succession up to three times. After each roll the user can decide to stay or roll again. If they decide to stay, they make a a certain amount of money (see table below). If they decide to roll again, they receive 0 dollars if their new roll is less than their previous one. They once again have the option of staying or rolling once more with the same outcomes. All we need is one die and a piece of paper with the amount of money they win given each outcome.

The die has 20 sides.
1-4: \$1
5-10: \$2
11-15: \$5
16-19: \$10
20: \$50

Empirical Data and Predictions

Empirical probability of loss: 33%
Empirical probability of 1: 0%
Empirical probability of 2: 3%
Empirical probability of 5: 17%
Empirical probability of 10: 40%
Empirical probability of 50: 7%

Empirical average winnings: \$8.4

We obtain these numbers by counting the number of times the player received a 0, 1, 2, etc and dividing each of these numbers by the number of times played (100). Thus, we could extrapolate the number of occurrences for each of these outcomes by dividing each of the percentages by 100 and (in this case, as there were 100 trials) multiplying by 100. So, we know, for example, that the player received 33 losses. The average winnings is obtained by multiplying the empirical probability of each outcome by the reward for that outcome. For example, we would multiply .33 by 0 and add it to .03*2, etc. This yields 8.4 dollars.

Theoretical Probabilities

Theoretical probability of loss: 61%
Theoretical probability of 1: .7%
Theoretical probability of 2: 5%
Theoretical probability of 5: 11.4%
Theoretical probability of 10: 15.6%
Theoretical probability of 50: 5%

Theoretical average winnings: \$4.7

We obtain these numbers by counting the number of outcomes that involve the player receiving a 0, 10, 20, etc and dividing each of these numbers by the number of possible outcomes (4620). Thus, we could extrapolate the number of occurrences for each of these outcomes by dividing each of the percentages by 100 and (in this case, as there were 4620 possible outcomes) multiplying by 4620. So, we know, for example, that the number of outcomes that leads to the player losing is 2850. The average winnings is obtained by multiplying the theoretical probability of each outcome by the reward for that outcome. For example, we would multiply .61 by 0 and add it to .05*20, etc. This yields 47 dollars. We assume, however, that the player has no skill and will stop rolling after receiving a 1 on the first roll even though there is no possible way the loser can lose on the following roll. This assumption accounts for a low empirical average. The empirical average observed is likely to be much higher than this. This is because we do not expect a user rolling once, receiving a 1, and stopping as likely as a user rolling three 1s (as we see it likely that the user will continue to try to roll as long as they canâ€™t lose). For this reason, we consider this theoretical probability a lower bound on the actual average winnings.

Expected Profits

We calculate sqrt(pqn) to find the standard deviation for each of the outcomes. N being 500.

Standard deviation of loss: 10.9
Standard deviation of 1: 1.86
Standard deviation of 2: 4.87
Standard deviation of 5: 7.1
Standard deviation of 10: 8.11
Standard deviation of 50: 4.87

Overall standard deviation: 16.9

Expected Costs and Revenue

The paper and pencil required to make the sign would cost around \$2 and the die would cost around \$1 which would bring out total expenses up to \$1+\$2=\$3 dollars. We would replace these monetary rewards with items such as stuffed animals which cost an equal amount to purchase. Which would allow us to bring the expected minimum winnings down to around \$5. We could then charge approximately \$8.5 dollars to play the game. This would seem like a reasonable fee for the prizes.

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