×
Back to all chapters

# Expected Value

If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.

# Expected Value - Problem Solving

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?

×