Monday, April 25, 2016

Published 4:32 PM by with 0 comment

Why Don't We Ever See Perfect Brackets In March Madness?

How unlikely is a perfect bracket?




Most brackets start with 64 teams. Each game involves two teams with one winner advancing. Thus, you need to get 32 picks right in the first round, 16 right in the second round, 8 in the third, 4 in the fourth, 2 in the fifth, and 1 in the sixth. Adding that up means you need to make 63 correct picks.

A very naïve way to approach this is to say that you pick blindly, so we start by assuming you have a 50% chance of the right pick in each game. That leaves us with:


This is extremely small. You are more likely to be both (not either or) crushed to death by a vending machine AND win the powerball.

That is very naïve though. Surely there are people out there with expert basketball knowledge who have a better than 50% chance of picking the winner of a game. Combining that with how unlikely it is that a 16 seed will beat a 1 seed, you can increase your odds. Let’s try a few of ways of handling this:
  • Assume the #1 and #2 seeds are guaranteed to win their first round games
  • Assume you're better than 50/50 overall when picking
  • Assume that your picking accuracy varies directly with the difference in team seeds

 

Assume the #1 and #2 seeds are guaranteed wins in the first round

This removes 8 picks that have to be correct. That gives us:


Not much better, and even this is a stretch as evidenced by the #2 seeds that have lost opening round games.

Assume you are better than 50/50 

Let’s say you’re far and away the best predictor of basketball games ever, and going in, you can pick each game with 70% accuracy (note this is before the tournament starts so in-tournament injuries can’t be factored in). That gives us:


This is actually not as unreasonable…it’s roughly one in 6 billion. If roughly every person on earth had this skill level for picking games and filled out a bracket, then we’d likely see a perfect bracket every year or so.

An interesting side calculation is what accuracy you would need to get a specific chance of a perfect bracket:


 


As a quick demo, what accuracy is required for a one in a million  chance? Plugging 1E-6 in for our chance yields an accuracy of 80%.

Varying chance by seed difference

Combining the effects to make it a little more complicated…say your accuracy varies linearly with seed difference with 100% for #1 vs #16 and 50% with a seed difference of 1 or 0. That gives you an accuracy of:



Also assume the better seed wins each game as your chances are best this way in this system. Your chance of getting the first round right for a given region is then 100%*92.9%*85.7%*78.6%*71.4%*64.3%*57.1%*50% which is roughly 8.2%. Since you have to get all four regions correct, the chance of getting the first round perfect in this system is:


This is not bad at all. You would expect to see perfect first rounds semi-regularly (it’s hard to handle situations like Michigan State this season so they aren’t guaranteed).

We now need to do the second round in this system. Same analysis as before means a given region has a roughly 13.1% chance of being perfect, and the chance of getting all four regions perfect is 0.00029. This looks better, but remember that we have to combine this with the first round chance, so we have to multiply the two numbers and we end up with:


Continuing for all rounds, we get a 0.6% chance for a perfect third round, 6% chance for a perfect fourth, 25% chance for a perfect fifth, and a 50% chance for a perfect sixth. Combining all of these yields:


This is pretty tiny…it’s worse than a 1 in a trillion chance. This is not as bad as picking completely blindly, but if everyone on earth had enough skill to do this and picked a bracket, you would still not expect this to have happened yet in the >=64 team tournament history.

You could easily perform a less naïve approximation of this to get different values (e.g., using betting odds or something like fivethirtyeight.com), but hopefully this illustrates how unlikely a perfect bracket actually is.

 




 

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