With the One Shining Moment and the rest of the 2018-19 basketball season now in the rear-view mirror, it is that time of year to reflect on the past season and (if you are me) crunch a few more numbers based on yet another year of accumulated stats. The first topic on my math to-do list is something that piqued my interested in the middle of this year's tournament (based on a question from the board). This year, it was reported that a single person on-line had successfully picked every single tournament game up to and including the Sweet 16. While this obviously seems like a tough thing to accomplish, the obvious question is "how tough?" What are the odds of this kind of bracket perfection, and what is the "correct" way to calculate it?
As you might expect, I am not the first person to think about this problem. Warren Buffet perhaps stimulated most of this discussion when in 2014 he started to offer multi-million or even $1 billion dollar rewards for different variations on a perfect bracket, either just up to the Sweet 16 or a full perfect bracket. Largely in response to this publicity, math and stats gurus started to take aim at answering the question as to how likely it was that Buffet would need to pay up.
There is one extremely simply way to think about this problem. If you assume a person is simply randomly guessing on the winner of each game, the odds are very easy to calculate. It is the same as the odds of correctly guessing the result of 63 consecutive coin flips, which is 1 in 2 multiplied by itself 63 times (2^63). In other words:
1 in 9,223,372,036,854,780,000
Those are pretty long odds. (Also note that although there are 68 teams in tournament, most mainstream bracket contests ignore the results of the First Four. With the remaining 64 teams, they play in 63 total games, with one team being eliminated in each game until 1 team remains).
Although a lot of people will reference the number above, it isn't correct. That is because games are clearly not just coin flips with each team having equal probability of victory. The obvious example of this is the set of first round games between 1-seeds and 16-seeds. If one were to enter a contest where one only needs to pick the winner of these 4 games, the obvious strategy is to take all the 1-seeds. In 40 of the past 41 tournament, this would be the winning strategy. However, in 2018, that would not have been the case, as UMBC upset 1-seed Virginia in historical fashion.
This particular example gives valuable insight into how to think about the problem of the odds of a perfect bracket. What would the odds be to "win" the 1-seed vs. 16-seed Challenge? In most years, the odds would be close to 100%, as long as one knew that 1-seeds (almost) always win in the 1st round. But, what about in 2018? Let's say that there was one brave UMBC grad that decided to take a flier on his alma mater. What would their odds have been? My math suggests that this type of upset should occur about 1% of the time (about once every 25 years). So, I think that it is reasonable to say that this UMBC grad had about a 1% chance to win this contest with that bracket.
This example tells us several things:
1) The odds to pick a "perfect bracket" seem to be equivalent to the odds of that bracket occurring
2) Therefore the odds of a perfect bracket are not the same from year to year and can vary widely
3) If you can estimate the probability of victory for each potential match-up, that should allow you to make the appropriate calculation
In my various internet searches related to this topic, most people who have thought about this problem have come to the same basic conclusions. Professors at various Universities have weighed in on the topic. One Professor at Depaul gave the odds at 1 in 128 billion. Another professor at Duke suggested it was 1 in 2.4 trillion. But, perhaps the best analysis I have seen is from Nate Silver at 538.com, who calculated the odds of a perfect bracket in 2014 to be 1 in 7.4 trillion, yet only 1 in 1.6 trillion in 2015.
As for my approach to this problem, I used essentially the same methodology as 538. I used the pre-tournament efficiency data from Kenpom to generate point spreads, and then used my own formula to calculate the probability that either team would win any potential match-up. When I ran these numbers for the 2019 tournament, I got the following numbers
Correct bracket up to the Sweet 16 (48 games): 1 in 550 million
Correct full bracket (all 63 games): 1 in 3.3 trillion
As for the man who picked the Sweet 16 correctly this year (who happens to be a Michigan fan from Saginaw...) his bracket was finally busted after Game 49. This was apparently the longest streak of correct picks on record. One analysis suggests the odds for this accomplishment were a bit better than I calculated (they said only 1 in 140 million based in this article) but their method used the probability extracted from the money line of each game, which I don't think is quite as accurate. (The odds derived from the money line for both teams in a single contest don't sum to one. That is partially how Vegas makes money...) So, in general, I think my numbers make sense.
But, I wanted to explore one final avenue. While we now know the odds of this specific tournament, I was wondering if I could get a feel for how much variation in these odds might exist. So, I used the same Kenpom efficiency and probability data to set up a simple "Monte Carlo"-style simulation of the 2019 tournament. In general, I wanted to use this method to try to help me fill out my office pool bracket (which was very, very helpful). Considering that I need the exact same probabilities to run the simulation as I do to calculate the odds of a specific bracket occurring in the first place, I could essentially kill both birds with one stone.
So, I ran a total of 5000 simulation of the 2019 and then calculated the distribution of the odds of a perfect bracket. The distribution of the odds is shown below, where a plot the log10 value of the odds. What I mean by that is "9" would be equal to 1 in a billion (which has 9 zeros). "10" is 1 in 10 billion, "11" is one in 100 billion, etc.
The shocking thing to me is how wide the spread is here. Over the 5000 simulations, the total odds for a perfect bracket varied by 11 orders of magnitude! Then again, in the example that I gave above, the single game upset of UMBC over Virginia in 2018 changed the odds of that bracket by a factor of roughly 100 (2 orders of magnitude) so maybe this distribution is not crazy after all...
80% of the odds fell between 1 in a trillion and 1 in 10 quadrillion. In the most reliable calculation examples shown above, most of the estimates are between 1 and 10 trillion, including for the 2014, 2015, and 2019 tournaments. But, based on the graph above, the majority of brackets should have odds that are in the 1 in 10 trillion to 1 in 1 quadrillion range. So, I am not sure if the brackets we have examined so far are just a little more "chalky" that other years, if there is something off in my calculations, or if in reality, the really good teams just beat the odds a little more often, resulting a less chaotic bracket than the math suggests. I do have some data that suggest 1-seeds win tournament games at a slightly higher rate than the Vegas spread suggests. Maybe there is a little truth in all three possible factors.
As for the extreme cases, the least chaotic bracket still had odds of around 1 in 30 billion. But, the craziest bracket in my particular simulation had odds of:
1 in 5,202,405,322,734,840,000,000 (or 1 in 5 sextillion!)
This value is roughly 500 times less likely than even the "random coin flip bracket" mentioned above. I think that might be a fun tournament to see play out, but we might need to wait 5000 years to see it.
That is all for now. Stay tuned for a bit more basketball math in the days ahead. I am not ready to turn my focus to football just yet.
As you might expect, I am not the first person to think about this problem. Warren Buffet perhaps stimulated most of this discussion when in 2014 he started to offer multi-million or even $1 billion dollar rewards for different variations on a perfect bracket, either just up to the Sweet 16 or a full perfect bracket. Largely in response to this publicity, math and stats gurus started to take aim at answering the question as to how likely it was that Buffet would need to pay up.
There is one extremely simply way to think about this problem. If you assume a person is simply randomly guessing on the winner of each game, the odds are very easy to calculate. It is the same as the odds of correctly guessing the result of 63 consecutive coin flips, which is 1 in 2 multiplied by itself 63 times (2^63). In other words:
1 in 9,223,372,036,854,780,000
Those are pretty long odds. (Also note that although there are 68 teams in tournament, most mainstream bracket contests ignore the results of the First Four. With the remaining 64 teams, they play in 63 total games, with one team being eliminated in each game until 1 team remains).
Although a lot of people will reference the number above, it isn't correct. That is because games are clearly not just coin flips with each team having equal probability of victory. The obvious example of this is the set of first round games between 1-seeds and 16-seeds. If one were to enter a contest where one only needs to pick the winner of these 4 games, the obvious strategy is to take all the 1-seeds. In 40 of the past 41 tournament, this would be the winning strategy. However, in 2018, that would not have been the case, as UMBC upset 1-seed Virginia in historical fashion.
This particular example gives valuable insight into how to think about the problem of the odds of a perfect bracket. What would the odds be to "win" the 1-seed vs. 16-seed Challenge? In most years, the odds would be close to 100%, as long as one knew that 1-seeds (almost) always win in the 1st round. But, what about in 2018? Let's say that there was one brave UMBC grad that decided to take a flier on his alma mater. What would their odds have been? My math suggests that this type of upset should occur about 1% of the time (about once every 25 years). So, I think that it is reasonable to say that this UMBC grad had about a 1% chance to win this contest with that bracket.
This example tells us several things:
1) The odds to pick a "perfect bracket" seem to be equivalent to the odds of that bracket occurring
2) Therefore the odds of a perfect bracket are not the same from year to year and can vary widely
3) If you can estimate the probability of victory for each potential match-up, that should allow you to make the appropriate calculation
In my various internet searches related to this topic, most people who have thought about this problem have come to the same basic conclusions. Professors at various Universities have weighed in on the topic. One Professor at Depaul gave the odds at 1 in 128 billion. Another professor at Duke suggested it was 1 in 2.4 trillion. But, perhaps the best analysis I have seen is from Nate Silver at 538.com, who calculated the odds of a perfect bracket in 2014 to be 1 in 7.4 trillion, yet only 1 in 1.6 trillion in 2015.
As for my approach to this problem, I used essentially the same methodology as 538. I used the pre-tournament efficiency data from Kenpom to generate point spreads, and then used my own formula to calculate the probability that either team would win any potential match-up. When I ran these numbers for the 2019 tournament, I got the following numbers
Correct bracket up to the Sweet 16 (48 games): 1 in 550 million
Correct full bracket (all 63 games): 1 in 3.3 trillion
As for the man who picked the Sweet 16 correctly this year (who happens to be a Michigan fan from Saginaw...) his bracket was finally busted after Game 49. This was apparently the longest streak of correct picks on record. One analysis suggests the odds for this accomplishment were a bit better than I calculated (they said only 1 in 140 million based in this article) but their method used the probability extracted from the money line of each game, which I don't think is quite as accurate. (The odds derived from the money line for both teams in a single contest don't sum to one. That is partially how Vegas makes money...) So, in general, I think my numbers make sense.
But, I wanted to explore one final avenue. While we now know the odds of this specific tournament, I was wondering if I could get a feel for how much variation in these odds might exist. So, I used the same Kenpom efficiency and probability data to set up a simple "Monte Carlo"-style simulation of the 2019 tournament. In general, I wanted to use this method to try to help me fill out my office pool bracket (which was very, very helpful). Considering that I need the exact same probabilities to run the simulation as I do to calculate the odds of a specific bracket occurring in the first place, I could essentially kill both birds with one stone.
So, I ran a total of 5000 simulation of the 2019 and then calculated the distribution of the odds of a perfect bracket. The distribution of the odds is shown below, where a plot the log10 value of the odds. What I mean by that is "9" would be equal to 1 in a billion (which has 9 zeros). "10" is 1 in 10 billion, "11" is one in 100 billion, etc.
The shocking thing to me is how wide the spread is here. Over the 5000 simulations, the total odds for a perfect bracket varied by 11 orders of magnitude! Then again, in the example that I gave above, the single game upset of UMBC over Virginia in 2018 changed the odds of that bracket by a factor of roughly 100 (2 orders of magnitude) so maybe this distribution is not crazy after all...
80% of the odds fell between 1 in a trillion and 1 in 10 quadrillion. In the most reliable calculation examples shown above, most of the estimates are between 1 and 10 trillion, including for the 2014, 2015, and 2019 tournaments. But, based on the graph above, the majority of brackets should have odds that are in the 1 in 10 trillion to 1 in 1 quadrillion range. So, I am not sure if the brackets we have examined so far are just a little more "chalky" that other years, if there is something off in my calculations, or if in reality, the really good teams just beat the odds a little more often, resulting a less chaotic bracket than the math suggests. I do have some data that suggest 1-seeds win tournament games at a slightly higher rate than the Vegas spread suggests. Maybe there is a little truth in all three possible factors.
As for the extreme cases, the least chaotic bracket still had odds of around 1 in 30 billion. But, the craziest bracket in my particular simulation had odds of:
1 in 5,202,405,322,734,840,000,000 (or 1 in 5 sextillion!)
This value is roughly 500 times less likely than even the "random coin flip bracket" mentioned above. I think that might be a fun tournament to see play out, but we might need to wait 5000 years to see it.
That is all for now. Stay tuned for a bit more basketball math in the days ahead. I am not ready to turn my focus to football just yet.
