# Coincidences: A Lesson in Expected Value

As I followed the election I noticed the frequent mentions counties (or cities) that have been known “predict” the presidential election winner. The idea is that a the winner of a certain county has matched the winner of the election for multiple elections. Let’s look at county A for an example. To simplify things lets assume the odds of predicting a winner in a presidential election are 50-50. This would mean that the probability of getting 8 elections right would be 1 in 256. This means that it is unlikely that county A would predict the election by chance. But what about the rest of the counties in America? There are over 3,000 counties in America (according to an economist article found here: http://www.economist.com/blogs/economist-explains/2016/11/economist-explains), so we can expect about 12 of these counties would have “predicted” the winner of the presidential election.

Rare events happen all the time. Rare are rare, but rare is not impossible. Let’s say that there is a (hypothetical) free sweepstakes with a 1 in 100 chance of winning \$100. You may not think that you wouldn’t know anyone who won, but if a sweepstakes like this exists you might be surprised about the likely outcome. It may not be likely that you specifically win, but if all your Facebook friends enter the contest someone you know is probably going to win. If you have at least 99 Facebook friends it is likely that you or someone you know will win the sweepstakes. You may think its a coincidence or luck, but it is really math. Expected value can’t tell you who is going to win, but it can tell you someone you know is likely to win. Now expected value is not a magic bullet. You may have 0 friends win or 2 friends win, but the most likely event is that someone will win. Unfortunately (legit) sweepstakes like this don’t exist, but it is a good example of how your perception of probability may not match reality. Another example is it probably going to rain 1 in 10 days where the probability of rain is 10%, but it is easy to pretend like it never rains when the probability of rain is 10%.

You may wonder why expected value matters. But it’s actually quite important when looking at everyday events. Sometimes it is easy to underestimate the chance that something odd or rare would happen. You may think it’s odd that runs when the meteorologist says the chance of that happening is 10%. Or that it only takes 23 people to have a 50% chance of there being, two people with the same birthday (details here). It is easy to forget that once in a lifetime event do happen once in a lifetime. How you think about probability is important. So before you yell at the TV meteorologist that said there was a 10% chance of rain but it rained, try to remember that unlikely does not equal impossible.

Coincidences: A Lesson in Expected Value by BrittanyAlexander is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International