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# Expected Value

Know what outcome to expect when you're dealing with randomness.

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The expected value of a random variable $$X$$ is the average value that $$X$$ takes. More formally, $E[X] = \sum_{x \in S} x \cdot P(X=x)$ where $$S$$ is the set of values that $$X$$ can take.

For example, the expected value of rolling a fair, six-sided die is $\frac{1}{6}(1)+\frac{1}{6}(2)+\frac{1}{6}(3)+\frac{1}{6}(4)+\frac{1}{6}(5)+\frac{1}{6}(6) = 3.5.$

There are $$12$$ blue marbles and $$4$$ red marbles in a bag. You reach into the bag and pull $$5$$ marbles at random. What is the expected value of the number of blue marbles drawn?

A fair coin is flipped until 2 heads are flipped in up to 3 successive flips. What is the expected value of the number of flips?

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