Bracket

How hard is it to pick a bracket for the basketball tournament?

Suppose it were blind; that is, if you had to choose the outcome without knowing the teams or their seeds. Then the likelihood of picking one game would be 50%, two games 25%, etc... There are 32 games in the first round, so that means you have to cut 1 in half 32 times: 1/2, 1/4, 1/8/ 1/16, 1/32, 1/64, 1/128, 1/256 etc... These are equivalent odds to tossing a perfectly balanced coin 32 times and having it come up heads.

Suppose you did that, then you would have to repeat the process for the 16 games in the second round, and so on. The problem is that if you've made some mistakes in the first round, you no longer have your exact teams in the second round: errors are compounded. If you had predicted the first round corretly, you would then have a 1/256 chance of predicting the results of the second round. So if you got 256 of your closest friends together, after the first round, and had them bet on the second, each one with a unique combination, one of you would win.

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Of course, you don't pick your teams blindly. Instead, you choose based on your estimations of the probability of winning. So a higher ranked team will, more often than not, beat a lower one. This improves the odds from 50% for most of the games. So for those 32 games in the first round, four were "upsets."

Now suppose you have really fine-grained knowledge of basketball, so you can guess better than the selection committee, and see where they have ranked a team lower than they might deserve. Then you can predict the upsets. As the rounds go on, you get teams that closer in rank playing each other, so you need to be an even better predictor, but here the odds shift slightly more in your favor, because there are fewer contests that you have to predict.

It looks like the seedings are slightly lower than a 90% predictor of outcomes in the first round (four out of 32), so you need to be better than that, picking the correct upsets. So far in the second round, there have been two upsets: Kansas losing to Stanford, Syracuse to Dayton. Stanford and Dayton were already supposed to have lost in the 1st round.