7.41 Consider a game in which a red die and a blue die are rolled. Let xR denote the value showing on the uppermost face of the red die, and define xB similarly for the blue die.

a. The probability distribution of xR is xR 1 2 3 4 5 6 p(xR) 1/6 1/6 1/6 1/6 1/6 1/6 Find the mean, variance, and standard deviation of xR.

b. What are the values of the mean, variance, and standard deviation of xB? (You should be able to answer this question without doing any additional calculations.)

c. Suppose that you are offered a choice of the following two games:

Game 1: Costs $7 to play, and you win y1 dollars, where y1 5 xR 1 xB.

Game 2: Doesn’t cost anything to play initially, but you “win” 3y2 dollars, where y2 5 xR 2 xB. If y2 is negative, you must pay that amount; if it is positive, you receive that amount.

For Game 1, the net amount won in a game is w1 5 y1 2 7 5 xR 1 xB 2 7. What are the mean and standard deviation of w1?

d. For Game 2, the net amount won in a game is w2 5 3y2 5 3(xR 2 xB). What are the mean and standard deviation of w2?

e. Based on your answers to Parts

(c) and (d), if you had to play, which game would you choose and why?