1.   After scoring a touchdown, a football team may elect to attempt a two-point conversion, by running or passing the ball into the end zone. If successful, the team scores two points. For a certain football team, the probability that this play is successful is 0.40.

a.   Let X = 1 if successful, X = 0 if not. Find the mean and variance of X.

b.   If the conversion is successful, the team scores 2 points; if not the team scores 0 points. Let Y be the number of points scored. Does Y have a Bernoulli distribution? If so, find the success probability. If not, explain why not.

c.    Find the mean and variance of Y.

2.    A certain brand of dinner ware set comes in three colors: red, white, and blue. Twenty percent of customers order the red set, 45% order the white, and 35% order the blue. Let X = 1 if a randomly chosen order is for a red set, let X = 0 otherwise; let Y = 1 if the order is for a white set, let Y = 0 otherwise; let Z = 1 if it is for either a red or white set, and let Z = 0 otherwise.

a.   Let pX denote the success probability for X.

b.  Find pX . Let pY denote the success probability for Y.

c.    Find pY . Let pZ denote the success probability for Z.

d.  Find pZ . Is it possible for both X and Y to equal 1?

e.  Does pZ = pX + pY?

f.      Does Z = X + Y? Explain.

3.    When a certain glaze is applied to a ceramic surface, the probability is 5% that there will be discoloration, 20% that there will be a crack, and 23% that there will be either discoloration or a crack, or both. Let X = 1 if there is discoloration, and let X = 0 otherwise. Let Y = 1 if there is a crack, and let Y = 0 otherwise. Let Z = 1 if there is either discoloration or a crack, or both, and let Z = 0 otherwise.

a.   Let pX denote the success probability for X. Find pX .

b.  Let pY denote the success probability for Y. Find pY

c.    Let pZ denote the success probability for Z. Find pZ .

d.  Is it possible for both X and Y to equal 1?

e.  Does pZ = pX + pY?

f.     Does Z = X + Y? Explain.