March is my favorite part of the year, and the NCAA March Madness is no small part of that. For those of you that are unfamiliar, after months of regular season basketball, weeks of conference tournaments, and two days of preliminary matches the field is narrowed to 64 teams. Those teams are put into a single-elimination bracket with four regions, and after 63 games a champion is crowned.
This year’s tournament has a little extra excitement as businessman Warren Buffett has offered $1,000,000,000 to anyone who can complete a perfect bracket. But Buffett didn’t luck into his fortune. Let’s use basic probability to determine just how likely it is that someone will be collecting that billion dollar check!
First of all, we need to determine whether we’re working with independent or dependent events. In this case, our outcomes are independent. The result of one game does not alter the odds in the next game. Dayton beating Ohio State doesn’t change the likelihood of Harvard beating Cincinnati. In contrast, a dependent event would be something like pulling marbles out of a bag without replacement. If we pull a blue marble out of the bag there is one less blue marble to pull out the next time, so the odds will change.
With independent events, we can compute the probability of all n events in a series happening by the following formula:
P(1)* P(2)* P(3)*…*P(n) = Probability of all events occurring
In this case, we need to know the probability of predicting all 63 events correctly. In our first case, let’s assume we know nothing about college basketball. In that case, we’ll randomly guess a winner from each matchup. Since we know nothing about the teams, we will estimate our likelihood of getting each guess correct at 1/2 or 0.5. In that case our likelihood of predicting a perfect bracket will be 1 in 2^63 or 1 in 9,223,372,036,854,775,808. That’s more than 9 billion billions! So without any knowledge it’s safe to say that your chances are not good.
But what if you know a little bit more? What if you know that a 1 seed has never lost to a 16 seed? That makes four games slam dunks and just leaves 59 left to figure out. Maybe you’re a college basketball expert and you can predict 60% of outcomes correctly. Your chances of getting a perfect bracket are now 1/(0.6)^59 or 1 in 12,275,963,663,147. We’re down to one in twelve million billion, but Mr. Buffett’s money is still looking pretty safe.
Let’s try one other example. Let’s say you’re a college basketball expert with a time machine that allows you to look two days into the future. That will allow you to know the first 32 results with absolute certainty, and just leave the final 31 games that you can predict correctly 60% of the time. Now your chances are 1/(0.6)^31 or 1 in 7,538,956. Even with expertise and a time machine your chances are just one in seven and a half million!
Three quick takeaways:
1. When events are independent, multiply the probabilities to find the total probability of all events happening
2. Exponential growth makes numbers get really big (or really small) really fast
3. Warren Buffett’s money is probably safe… until we build a better time machine!