Monday, March 31, 2014

NCAA bracket odds

We're down to the "Elite 8" in the 2014 NCAA tournament. I've been hearing some discussion of the odds of filling a "perfect bracket." It has been discussed at On Point Radio, which I listen to frequently. The program aired on March 26, 2014. USA Today has a youtube video up of a DePaul math professor estimating the odds. He comes up with an estimate that a savvy bracket filler has about 1 in 128 billion odds of filling in a perfect bracket. It's March Madness.

Then I thought to myself, how did the professor get there? USA today does not go into the details. I wonder what the odds are of finding real math in USA today on a given today? Probably slimmer than 1 in 1 million. I'll point out that precision does not matter when we're finding astronomically long odds. I'll call this a ballpark estimate. With probabilities this small, the upshot is that you'll never win. It's worse than playing lotto.

Let's do some math. Sixty-four teams compete in the NCAA tournament. There are six rounds, including the final game. So we have 32+16+8+4+2+1 = 63 games played. In order to fill out a perfect bracket, I have to choose 63 winners correctly. If we were flipping a coin, there are

possible sequences of 63 coin flips. A bracket filler would have odds of about 1 in 9 quintillion of filling out the bracket correctly. There is no hope of winning here.

But we are not flipping a coin. Suppose you're a savvy odds maker. You know which teams are ranked highest in the tournament. You study basketball statistics, etc. I will quantify your savvy thusly: you have a 90% probability of picking a winner of 16 games in the first round. You have a 90% probability of picking a winner of 16 games in the second round. Your probability of picking all the other winners is 50/50, the same as flipping a fair coin. What is your probability of filling in the bracket correctly? You have to correctly choose 63 mutually exclusive events with a probability of .9 for 32 of them, and .5 for the other 31. Since the events are mutually independent, the individual probabilities multiply. That gives us a final probability of

or odds of

This is roughly twice as much probability as the DePaul math professor estimated. So, my hypothetical savvy guesser had roughly twice as much savvy, but he is still hopeless when it comes to filling in the bracket correctly.

Here is some R code to calculate and print the numbers here. We're right on the edge of R's basic precision, where p could be rounded down to zero. If I change the numbers slightly, I would need to use the Rmpfr package (or logarithms) to get a non-zero printout.

> p=c(0.9,0.5)
> games=c(32,31)
> prob=prod(p^games)
> prob
[1] 1.598934e-11
> 1/prob
[1] 62541682939
> format(1/prob,dig=3,sci=T)
[1] "6.25e+10"


  1. I just had this discussion with my family before March Madness began, because of the contest that offered a billion dollars to someone who could pick a perfect bracket. We talked about probability and odds-makers, etc. My husband actually filled out a couple brackets just for fun (they're destroyed).

    One year I participated in the office pool and came out second for one reason and one reason only. Higher point values were awarded for picking winners in the upper rounds. I was the only person who picked Wisconsin to go all the way (just because I'm an alumna, I know nothing about basketball), and they weren't very good that year. They made into the final four, if I remember correctly. Because of that, no one else's bracket was surviving, and I got lots of points that no one else got because WI went so far in the bracket. :-) Sheer luck and I get the bragging rights. Ha!

  2. Yes, when competing against others in the office pool, your probability of winning is much better than that of filling in a perfect bracket.

  3. interesting blog - i am a huge college basketball fan (go KU!), and fill out a bracket every year. I am still waiting for that perfect bracket! looks like it may take a while.