You can have the best analytics on your side. You could even watch 100 hours of college basketball a week to prepare for filling out your bracket. None of this matters for the wrong pool. And just about the only wrong sort of pool is a big one. Let me repeat: DO NOT enter a big pool. Here's why.

## Drunk people throwing darts at a dartboard

You and a friend walk into a bar and find the employees of a start-up company there. They have just secured their Series A funding and feel good about their future. In anticipation of becoming millionaires, they start throwing back flutes of Champagne. After a few too many drinks, and against better judgment, they decide to play a game.

Each person gets one throw at a dartboard. Hit the bullseye, earn a free drink.

You laugh at the spectacle, thinking that not a single person will hit the bullseye. It even doesn't matter that the bar has installed a magnetic field that directs all errant darts back towards the dartboard—great for safety, but the bar still won't be serving up any drinks.

Your friend bets you that someone will hit the bullseye. Should you take the bet?

It depends on the number of people lined up to throw a dart. Let's assume that a dart from a drunk person has an equal chance of landing anywhere on the dartboard thanks to the magnetic field. The odds that any one drunk hits the bullseye is small, about 0.5 percent.

However, to win your bet, you need every drunk person to miss. There is a 99.5 percent probability that the first drunk misses, but you must multiply 0.995 by 0.995 to get the likelihood that both the first and second drunk miss. If the company has 20 drunks that step up to fling a dart, there's a 90.5 percent chance that all of them miss. This implies a 9.5 percent chance that at least one drunk hits the bullseye.

For an increasing number of drunks, the probability that at least one hits the bullseye increases rapidly. At 100, there's a 39.4 percent chance for someone to hit the bullseye, and this probability increases to 86.5 percent for 400 people.

The same principle applies to your March Madness pool. Suppose you're filling out a pool in 2010. Kansas has just capped an amazing regular season and heads into the tourney as a 1 seed. Analytics agrees with this assessment, as the Jayhawks top The Power Rank heading into the tourney.

However, everyone in your pool has also picked Kansas. According to data from ESPN, 41.8 percent of brackets filled out on their site had Kansas as champion. If Kansas wins, you and many others get those 32 points.

However, just like the drunk people throwing darts, someone else in your pool will hit the bullseye in the earlier rounds. They will get lucky and pick two surprise Sweet 16 teams or a shocking Elite 8 team.

Only one person has to get lucky on some weird pick or another to topple you from the top of your pool. This gets more likely with more people in your pool.

## How your chance to win depends on pool size

Let's put some numbers behind how your chance to win a pool depends on its size.

Suppose you fill out the bracket with all favorites. Since 2002, the higher ranked team in The Power Rank has won 71.3 percent of 844 tournament games. This suggest picking the higher ranked team for each game.

Most years, the bracket will look boring, and you might stab yourself in the eye from having to repeatedly cheer for teams like Duke and Kentucky. But you want to win a pool, right?

To determine how often this bracket of favorites wins a pool, I developed a simulation method that accounts for two types of randomness in your pool. Researchers use these types of "Monte Carlo" simulations to study phenomena ranging from polymeric materials to the stock market.

First, the simulation must account for the inherent randomness in playing the game of basketball. In real life, the tournament only happens one time. In 2010, Kansas fell in the Round of 32 to Northern Iowa, and Duke went on to win the tourney.

However, if the same tourney happened again, the results would be different. Northern Iowa's Ali Faroukmanesh misses that three point shot, and Kansas survives and advances. Then maybe they beat Duke in the Final Four and win the tourney.

To handle this variance in basketball, the simulation uses my win probabilities at The Power Rank. For each game, a coin is flipped according to this win probability.

For example, if Kansas has a 97 percent chance to win their first game, this coin comes up heads on 97 percent of flips. Kansas advances on each heads. We simply repeat this procedure for each game in each round to simulate the results of the tourney.

### Randomness in a pool

Second, there is variability among the brackets in your pool. A person in your pool might not know anything about college basketball and pick games depending on which mascot he likes.

Another person went to Connecticut as an undergrad and always picks the Huskies as champion. After Connecticut won 2 of 4 tourneys from 2011 through 2014 as a long shot, this person might not get invited back into the pool.

For the variability in brackets in your pool, we consult the data on brackets submitted to ESPN. To simulate a 50 person pool, the computer pulls 50 brackets at random from the millions submitted on ESPN.

This is clearly an approximation to the actual brackets that might appear in your pool. For example, if you live in North Carolina, your pool will probably have more brackets with Duke and North Carolina as champions than the national average.

## How your chances depend on pool size

Then I performed simulations at a number of different pool sizes and tracked the fraction of simulations in which a bracket of favorites won. The visual shows how your chance to win a pool decreases rapidly with pool size for the 2010 tourney.

By picking all favorites, you have a 43 percent chance to win a 5 person pool. You'll win about every other year, which is pretty good. If winning is your only goal, enter a small pool.

For a 30 person pool, your chance of winning the pool drops to 10.3 percent. With your choice of Kansas as champion, there are enough others in your pool with this choice that someone else will win based on luck in picking earlier games.

In terms of investment potential, a win probability of 10.3 percent for a 30 person pool is pretty good. If you took a random bracket from ESPN and submitted it, you would have same chance as anyone else, or 3.3 percent, to win the pool. By submitting the favorites based on my numbers, your average return on this investment is over 300 percent. The public is like those drunk people throwing darts.