Heads Minus Tails Distribution

A fair Coin is tossed $2n$ times. Let $D\equiv\vert H-T\vert$ be the absolute difference in the number of heads and tails obtained. Then the probability distribution is given by

$$P(D=2k)=\cases{ ({\textstyle{1\over 2}})^{2n}{2n\choose n} ... ...xtstyle{1\over 2}})^{2n}{2n\choose n+k} & $k=1$, 2, \dots,\cr}$$

where $P(D=2k-1)=0$. The most probable value of $D$ is $D=2$, and the expectation value is

$$\left\langle{D}\right\rangle{}={n{2n\choose n}\over 2^{2n-1}}.$$

See also Bernoulli Distribution, Coin, Coin Tossing

References

Handelsman, M. B. Solution to Problem 436, ``Distributing `Heads' Minus `Tails.''' College Math. J. 22, 444-446, 1991.