Expected Value

What is the expected number of coin flips we must make, before we see a Head followed immediately by a Tail?

Let \(X, \ Y, \ Z\) be random variables over \(\mathbb{Z}_{3}, \ \mathbb{Z}_{3}, \ \mathbb{Z}_{4},\) respectively, where \(\mathbb{Z}_n\) denotes \(\{1, 2, \dots, n\}.\) If \(P_{X, Y, Z}(x, y, z) = \frac{xy}{144},\) what is the greatest integer less than or equal to \(E[X + Y | Z]?\)

Suppose \(E[X|Y=0] = 1\) and \(E[X|Y=1] = 9.\) When \(P(Y=0) = 0.2\) and \(P(Y=1) = 0.8,\) what is \(E[X]?\)

Given \(Z=9,\) if \(X\) and \(Y\) are uniform random variables over \(\{0, 1, \dots, 9\},\) what is \( E[ X + Y | Z=9 ] ?\)

When flipping a fair coin repeatedly, what is the expected value of the number of trials needed to get two tails in a row?

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