(Solution) - A penny and a nickel are tossed Both are fair -(2025 Original AI-Free Solution)

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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.
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. Are X and Y independent?
e. Does pZ = pX pY?
f. Does Z = XY? Explain.