Early Election Results

Let Jones and Smith be the only two contestants in an election that will end in a deadlock when all votes for Jones ($J$) and Smith ($S$) are counted. What is the Expectation Value of $X_k\equiv \vert S-J\vert$ after $k$ votes are counted? The solution is

$$\left\langle{X_k}\right\rangle{} = {2N{N-1\choose \left\lfloor{k/2}\right\rfloor }{N-1\choose \left\lfloor{k/2}\right\rfloor -1}\over{2N\choose k}}$$

$$= \left\{\begin{array}{ll} {k(2N-k)\over 2N} {N\choose k/2}^2{2N\ch... ...\choose (k-1)/2}^2{2N\choose k-1}^{-1} & \mbox{for $k$\ odd.}\end{array}\right.$$

References

Handelsman, M. B. Solution to Problem 10248. ``Early Returns in a Tied Election.'' Amer. Math. Monthly 102, 554-556, 1995.