Cooper Doyle

2022-07-08

If you are a voter in a state with a million other active voters, what is the probability that your vote will be pivotal in the next election?
Assuming a given voter is equally likely to vote for either party, it seems that the answer is $\left(\begin{array}{c}1,000,000\\ 500,000\end{array}\right)\ast \left(\frac{1}{2}{\right)}^{500,000}\ast \left(\frac{1}{2}{\right)}^{500,000}$ and when I plug it into a calculator it gives me 7.9788E-4, which doesn't seem plausible. Is there something wrong with my reasoning?

pampatsha

Expert

The probability that your vote will be pivotal in deciding this kind of perfect election can be calculated as the probability that 500,000 people will vote for one candidate and 5000,000 people will vote for the second candidate. Just as you calculated, this probability will be $\left(\genfrac{}{}{0}{}{1,000,000}{500,000}\right)×\left(\frac{1}{2}{\right)}^{1000000}=7.9788...×{10}^{-4}.$ This probability is somewhere around one in a thousand. You might have been expecting a probability closer to one in a million, but the probability is as high as it is because of the law of large numbers. The great majority of the time the final count for each side will fall somewhere in between $500,000±1000.$ In other words, the race will tend to be fairly close, so there is a relatively high probability that your vote (and everyone else's) will sway the election.