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?