A penny and a nickel are tossed. Both are fair coins. Let X = 1 if the penny comes up heads, and let X = 0 otherwise. Let Y = 1 if the nickel comes up heads, and let Y = 0 otherwise. Let Z = 1 if both the penny and nickel come up heads, and let Z = 0 otherwise.

Question

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

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

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

d. Are X and Y independent?

e. Does

f. Does Z = XY? Explain.

Answer 1

(a) = 1/2

(b) = 1/2

(c) = 1/4

(d) Yes.P(X = x and Y = y) = P(X = x)P(Y = y) for all values of x and y.

(e) Yes.= 1/4 = (1/2)(1/2) = .

(f) Yes. If both coins come up heads, then X =1, Y =1, and Z = 1, so Z = XY. If not, then Z = 0, and either X, Y, or both are equal to 0 as well, so again Z = XY.