# Pairwise Independence $$\neq$$ Mutual Independence

Let $$X$$ and $$Y$$ be random variables describing independent tosses of a fair coin. Let $$Z$$ be the random variable that equals $$1$$ if both $$X$$ and $$Y$$ land heads and that equals 0 otherwise.

How many of the following statements are true?

• The collection of random variables $$\{X, Y\}$$ is independent.
• The collection of random variables $$\{X, Z\}$$ is independent.
• The collection of random variables $$\{Y, Z\}$$ is independent.
• The collection of random variables $$\{X, Y, Z\}$$ is independent.
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