Two dice are rolled. Let X = 1 if the dice come up doubles and let X = 0 otherwise. Let Y = 1 if the sum is 6, and let Y = 0 otherwise. Let Z = 1 if the dice come up both doubles and with a sum of 6 (that is, double 3), and let Z = 0 otherwise.
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.
(a) = 1/6
(b) = 5/36
(c) = 1/36
(d) No. 
(e) No. 
.
(f) Yes. If the dice come up double 3, thenX=1, Y =1, andZ= 1, soZ =XY. Ifnot, then Z= 0, and eitherX, Y, or both are equal to 0 as well, so again Z = XY.
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