Reconsider Prob. 22.4-8 involving the game of craps. Now the objective is to estimate the probability of winning a play of this game. If the probability is greater than 0.5, you will want to go to Las Vegas to play the game numerous times until you eventually win a considerable amount of money. However, if the probability is less than 0.5, you will stay home. You have decided to perform simulation on a spreadsheet to estimate this probability. Perform the number of iterations (plays of the game) indicated below twice.
(a) 100 iterations.
(b) 1,000 iterations.
(c) 10,000 iterations.
(d) The true probability is 0.493. What conclusion do you draw from the above simulation runs about the number of iterations that appears to be needed to give reasonable assurance of obtaining an estimate that is within 0.007 of the true probability?
Prob. 22.4-8
The game of craps requires the player to throw two dice one or more times until a decision has been reached as to whether he (or she) wins or loses. He wins if the first throw results in a sum of 7 or 11 or, alternatively, if the first sum is 4, 5, 6, 8, 9, or 10 and the same sum reappears before a sum of 7 has appeared. Conversely, he loses if the first throw results in a sum of 2, 3, or 12 or, alternatively, if the first sum is 4, 5, 6, 8, 9, or 10 and a sum of 7 appears before the first sum reappears.
(a) Formulate a spreadsheet model for performing a simulation of the throw of two dice. Perform one replication.
(b) Perform 25 replications of this simulation.
(c) Trace through these 25 replications to determine both the number of times the simulated player would have won the game of craps and the number of losses when each play starts with the next throw after the previous play ends. Use this information to calculate a preliminary estimate of the probability of winning a single play of the game.
(d) For a large number of plays of the game, the proportion of wins has approximately a normal distribution with mean = 0.493 and standard deviation = 0.5√n. Use this information to calculate the number of simulated plays that would be required to have a probability of at least 0.95 that the proportion of wins will be less than 0.5. 22