TITLE: Expected Value of Number of Tails Minus Number of Heads
QUESTION [5 upvotes]: I have the following problem where, given $X_n$ is a random variable that equals the number of tails minus the number of heads when n fair coins are flipped, what is the expected value of $X_n$?
I am having a difficulty getting started on this problem.  Could someone offer a suggestion as to how this problem should be modeled?

REPLY [5 votes]: We can start and finish by saying that by symmetry the expected number is $0$. 
Or let $X$ be the number of tails. So $n-X$ is the number of heads. Thus tails minus heads is $2X-n$. This has mean $2E(X)-n$, which is $0$. This second argument will also give the answer for a coin that is biased, having probability $p$ of tails. Then the expected value of the difference is $2pn-n$.