The game of craps is played by rolling two balanced dice. A first roll of a sum of 7 or 11 wins; and a first roll of a sum of 2, 3, or 12 loses. To win with any other first sum, that sum must be repeated before a sum of 7 is thrown. It can be shown that the probability is 0.493 that a player wins a game of craps. Suppose we consider a win by a player to be a success, s.

a. Identify the success probability, p.

b. Construct a table showing the possible win-lose results and their probabilities for three games of craps. Round each probability to three decimal places.

c. Draw a tree diagram for part (b).

d. List the outcomes in which the player wins exactly two out of three times.

e. Determine the probability of each of the outcomes in part (d). Explain why those probabilities are equal.

f. Find the probability that the player wins exactly two out of three times.

g. Without using the binomial probability formula, obtain the probability distribution of the random variable Y, the number of times out of three that the player wins.

h. Identify the probability distribution in part (g).