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.