# Math problem alert: NCAA Magic or Unsurprising?April 6, 2016 12:04 PM   Subscribe

We had a NCAA March Madness contest at work, where about 40 of us all filled out brackets on ESPN's site. One of us - the winner of our group - did very well and ended up ranked 6000th out of all 13 million ESPN brackets - top .05%. My question is, was this really statistically unlikely or, given that there were 40 of us, is it unsurprising one of us did that well (e.g., the birthday problem)? What mathematical tools would one use evaluate this question? Thanks!
posted by slide to Grab Bag (9 answers total)

If we assume that an individual is equally likely to hit any given place of those 13 million places, there's a 6,000/13,000,000 = 6/13,000 = 4.6 * 10^-4 chance that any one of you would be in the top 6,000 entries. You then multiply that by 40 (since there are 40 independent chances) to get the final probability. That's 4.6 * 10^-4 * 40 = ~.018 or about 1.8% chance of this happening.
posted by Special Agent Dale Cooper at 12:12 PM on April 6, 2016 [1 favorite]

The probability of one bracket being in the top 6000 is 6000 / 13000000. The probability of at least one of forty of you doing it is forty times that. It works out to about 1.8%. So unlikely, but not lotto unlikely. You would expect about 1 of 54 companies to have this result, assuming similar participation at all.

If there are fewer than 13 million possible brackets, then a lot of the brackets will be duplicates and the odds will go up accordingly. I don't know enough about the tournament to know whether this is the case.
posted by kindall at 12:15 PM on April 6, 2016

If there are fewer than 13 million possible brackets

It's a couple more than that actually.
posted by zachlipton at 12:20 PM on April 6, 2016 [1 favorite]

Contrary to the first two answers in this thread, you don't multiply by 40, but because the probabilities are very small, it's a decent approximation anyway.

Here's the exact math. The chance that any given person got in the top 0.05% is 0.0005. So the chance that any given person did NOT get in the top 0.05% is 1 - 0.0005 = 0.9995. The chance that all 40 people did not get in the top 0.05% is 0.9995^40 = .9802. So the chance that at least one of them DID is 1 - .9802 = 0.0198 = 1.98%.

The reason that the 1.8% number other people are getting is pretty close to the correct answer is that they're calculating nx instead of 1 - (1-x)^n (here n = 40, x = 0.0005), but when nx is small, then those two quantities are close (this has to do with the binomial theorem).

You can tell that just multiplying by 40 can't be right because it implies that if you have more than 2000 people, the chance that someone would be in the top 0.05% would be greater than 1.
posted by dfan at 12:42 PM on April 6, 2016 [17 favorites]

(Actually, the numbers should be even closer because I was using a straight 0.05% and they were using 6,000/13,000,000 = 0.046%. In that case the exact method gives 18.3% and the approximate method gives 18.5%. It makes sense that the exact method is smaller because we know that the approximate method gives answers that are too large as the number of people in your company goes up.)
posted by dfan at 12:47 PM on April 6, 2016 [1 favorite]

All right, much simpler than I had expected! Hard to believe I was a math major once upon a time. Thanks everyone for the explanation.
posted by slide at 1:12 PM on April 6, 2016

But your friend has knowledge and expertise about college basketball. This makes his success more likely than the calculations would suggest. The math assumes that everyone is identical.
posted by JimN2TAW at 1:51 PM on April 6, 2016

But your friend has knowledge and expertise about college basketball. This makes his success more likely than the calculations would suggest. The math assumes that everyone is identical.

There's a lot more than 13,000,000 possible brackets. 13,000,000 presumably is how many brackets people actually submitted to ESPN. There's already a selection bias in place here - people who know less about college basketball are less likely to make brackets. But depending on the OP's job, it might be true that their coworkers could be atypically good at this sort of thing. However, certainly chance plays a big factor. Even if you knew the exact probabilities for each matchup, there's 60+ games, and some underdogs will almost certainly win.
posted by aubilenon at 2:38 PM on April 6, 2016

We're lawyers, and the stellar performer was a woman - and a sports fan but not sports crazy. I doubt she watched more than 1-2 games of the tournament.
posted by slide at 3:28 PM on April 6, 2016 [1 favorite]

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