easy Probability
A release-checked easy problem for training Discrete Random Variables, Expected Value.
Billy flips a fair coin $n$ times. Let $X$ denote the number of heads, and let $Y$ denote the number of tails. $\mathbb{E}[XY]$ can be expressed in the form \[\begin{aligned} \frac{an^2 + bn + c}{d} \end{aligned}\] for some integers $a, b, c, d$, where $a, b, c$ are each relatively prime with $d$. Compute $a + b + c + d$.
This Probability problem is tagged Discrete Random Variables, Expected Value. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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