# Will you be the deciding vote?

Another timely election-related Riddler problem. What are the odds of being the deciding vote?

You are the only sane voter in a state with two candidates running for Senate. There are N other people in the state, and each of them votes completely randomly! Those voters all act independently and have a 50-50 chance of voting for either candidate. What are the odds that your vote changes the outcome of the election toward your preferred candidate?

More importantly, how do these odds scale with the number of people in the state? For example, if twice as many people lived in the state, how much would your chances of swinging the election change?

Here is my solution:
[Show Solution]

## 3 thoughts on “Will you be the deciding vote?”

1. Dmytro Taranovsky says:

And if N is odd, $p(N) = \frac{1}{2^N} {N \choose \frac{N-1}2 }$ regardless of how the ties are handled.

1. Yes, and we can find the first subleading correction, giving
p(N) = sqrt(2/pi N) ( 1 – 1/4N ) for N even, (1 – 3/4N) for N odd
which improves the fit for small N (and invisibly for large N).