# Sports Odds versus Probabilities

NPR this morning announced (listen here) that Quicken Loans will pay \$1 billion for anyone who picks a perfect March Madness bracket. The odds of this are very long, but not 9 quintillion to 1, as reported.

NPR seems to have come up with the 9 quintillion to one figure—9 followed by 18 zeros—by calculating the total number of head-to-head matchups that are possible; see orgtheory.net for the exact figure. But that is, strictly speaking, the probability of picking the correct bracket if all matchups are equally possible. They are not: we have a high degree of confidence that the second round will have zero 16-seeds playing, because 16-seeds have never beaten 1-seeds in the first round. So we should put very little weight on the probability that any bracket with a 16-seed surviving the first round is the correct bracket.

This is “sports odds,” which take into account the probability of victory for each head-to-head matchup in an iterative fashion (if the probability of 16 beating 1 is low, then the probability of any bracket with 16 surviving is low, and even within those low-probability brackets in which 16 does survive, the probability of the 16 surviving the second round is low, meaning that the probability of any Sweet 16 round with a 16 even lower, etc.). So long as the probability of victory is not equal across all potential teams and all potential matchups, the odds of picking the right bracket are not 9 quintillion to 1. The exact figure is obviously a lot harder to calculate, and the odds are certainly still extremely long against picking the exact bracket, but this is the proper way to think about the problem.