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- 1. Double Down Youcef Azzi Jung Kim
- 2. Basic Rules: ● If you roll a double (1 and 1, 2 and 2, etc.) you win 0 dollars. ● If the roll isn’t a double, you win that sum (die one plus die two’s face number) ● The price to roll once is 7 dollars. ● You can roll as many times as you want as long as you pay the price each time.
- 3. Expected Value and Standard Deviation Theoretical for X. (X=money earned) ● Variance of X= 6/36(0-5.83)^2 + 2/36(3-5.83)^2 + 2/36(4-5.83)^2 + 4/36(5-5.83)^2 + 4/36 (6-5.83)^2 + 6/36(7-5.83) + 4/36(8-5.83)^2 + 4/36(9-5.83)^2 + 2/36(10-5.83)^2 + 2/36(115.83)^2 = 10.694 ● Standard Deviation = 3.27 dollars ● Mean of X= (0)(6/36) + (3)(2/36) + (4)(2/36) + (5)(4/36) + (6)(4/36) + (7)(6/36) + (8)(4/36) + (9)(4/36) + (10)(2/36) + (11)(2/36) ● X 0 3 Mean of X = 5.83 dollars 4 5 6 7 8 9 10 11 P(X) 2/36 4/36 4/36 6/36 4/36 4/36 2/36 2/36 6/36 2/36
- 4. Expected Value and Standard Deviation Experimental for X. (X=money earned) ● Variance of X = 11/36(0-7.56)^2 + 3/36(3-7.56)^2 + 4/36(4-7.56)^2 + 3/36(57.56)^2 + 7/36(6-7.56)^2 + 5/36(7-7.56)^2 + 3/36(8-7.56)^2 + 5/36(9-7.56)^2 + 2/36(10-7.56)^2 + 6/36(11-7.56)^2 = 24.29 ● Standard Deviation = 4.93 ● Mean of X = (0)(11/36) + (3)(3/36) + (4)(4/36) + (5)(3/36) + (6)(7/36) + (7)(5/36) + (8)(3/36) + (9)(5/36) + (10)(2/36) + (11)(6/36) ● Mean of X = 7.56 dollars X 0 3 4 5 6 7 8 9 10 11 P(X) 11/36 3/36 4/36 3/36 7/36 5/36 3/36 5/36 2/36 6/36
- 5. Simulation
- 6. Conclusion After setting up our game and calculating the average winnings for a player we set the price for one roll to be 7 dollars as the theoretical winnings of a player was 5.83 dollars. We did so in order to make sure we had a “house advantage”. Unfortunately, after playing the game 50 times, the average winnings for that set of trials, made the house lose money, as the average winnings is 7.56 dollars, which is more than the cost to play. Instead of the house gaining $0.17 each game, there was a loss of $0.56 per game as shown in our trials outcome. However, we are fairly certain, that if the game play was played 1,000 or more times the average experimental winnings would fall closer to the average theoretical mean, therefore making the house win money in the long run. The standard deviation for the theoretical calculations was fairly low, which is good, as we want consistency in order to win money. The standard deviation for the experimental calculations was fairly high, which is not ideal as we want consistency in order to insure profit in most of the games. If we could change the game in order to create more revenue and still keep the game fun, we would raise the price to play to $8.