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Last Updated: April 9, 2019

# Probability of a Perfect Bracket

## Introduction

A question I get a lot in March is, "What is the probability of filling out a March Madness bracket completely perfectly?" A frequently quoted statistic in the media is 1 in 9,223,372,036,854,780,000. This quotation is based on guessing randomly, which nobody does, and makes the hair on the back of my neck stand up. Here I will give a basic strategy you can follow, knowing nothing about basketball, and an estimate of your probability of success with said strategy.

## Rules

Following are the rules of the tournament. If you already know them, you can skip this part.

For most intents and purposes, the tournament is single elimination, starting with 64 teams. To satisfy my perfectionist readers, there are actually 68 teams, where eight of them are matched up in four games. The winners of those four games, and 60 other teams, result in 64 teams, which is when the bracket picking typically begins.

The 64 teams are divided into four different conferences, North, South, West, and East, of 16 teams each. Each team in each conference will have a "seed," from 1 to 16, which is meant to be a ranking of their ability, although it is an imperfect method. Next the following will happen:

1. The four 1 seeds will play the four 16 seeds.
2. The four 8 seeds will play the four 9 seeds.
3. The four 5 seeds will play the four 12 seeds.
4. The four 4 seeds will play the four 13 seeds.
5. The four 6 seeds will play the four 11 seeds.
6. The four 3 seeds will play the four 14 seeds.
7. The four 2 seeds will play the four 15 seeds.
8. The four 7 seeds will play the four 10 seeds.
9. After the games in steps 1 to 8, the 64 teams will be narrowed down to 32.
10. The winners from rule 1 will play the winners from rule 2.
11. The winners from rule 3 will play the winners from rule 4.
12. The winners from rule 5 will play the winners from rule 6.
13. The winners from rule 7 will play the winners from rule 8.
14. After the games in steps 10 to 13, the 32 remaining teams will be narrowed down to 16.
15. The winners from rule 10 will play the winners from rule 11.
16. The winners from rule 12 will play the winners from rule 13.
17. After the games in steps 15 and 16, the 16 remaining teams will be narrowed down to eight.
18. The winners from rule 15 will play the winners from rule 16.
19. After the games in step 18, the eight remaining teams will be narrowed down to 4. This will establish a winner from each of the four divisions.
20. The winner of the regional that had the #1 overall team as determined by the NCAA basketball committee when selecting the tournament teams plays the winner of the regional that had the #4 overall team; with the winner advancing.
21. The winners of the two regions, not mentioned in rule 20, play each other, with the winner advancing.
22. The winners of steps 20 and 21 will play, the winner winning the tournament.

Once it gets to 64 teams, there are 32+16+8+4+2+1=63 games in the tournament.

## Basic Strategy

Always pick the team with the higher-ranking seed, or lower number. For example, in the four games where a number 1 seed plays a number 16 seed, pick the number 1 seed. Of the 136 times a 1 seed has played a 16 seed, the number one seed won 135 times. 2018 saw the first number one seed lose to a 16 seed when UMBC (which has a great chess team) beat the University of Virginia.

When you get down to a number one seed winning each of the four divisions, pick the remaining three games randomly.

## Analysis

The following analysis is based on the 34 seasons from the inception of the tournament in 1985 through 2019. The following table shows the number of times each unbalanced match-up has occurred, the number of times the higher ranking seed won, and the probability, based on this data, of the higher ranking team winning. Situations where a match-up has never happened, like a one seed playing a 15 seed, are not shown.

Higher
Ranking
Seed
Lower
Ranking
Seed
Games
Played
Higher
Seed
Wins
Probability
Higher
Seed Wins
1 1 beats 16 0.992857 4 0.971733
1 2 beats 15 0.935714 4 0.766607
1 3 beats 14 0.850000 4 0.522006
1 4 beats 13 0.792857 4 0.395166
1 5 beats 12 0.642857 4 0.170788
1 6 beats 11 0.628571 4 0.156106
1 7 beats 10 0.607143 4 0.135883
1 8 beats 9 0.507143 4 0.066149
2 1 beats 8 0.805556 4 0.421097
2 4 beats 5 0.554054 4 0.094234
2 3 beats 6 0.602740 4 0.131983
2 2 beats 7 0.703704 4 0.245222
3 1 beats 4 0.730159 4 0.284229
3 2 beats 3 0.607143 4 0.135883
4 1 beats 2 0.530303 4 0.079085
5 1 beats 1 0.500000 2 0.250000
6 1 beats 1 0.500000 1 0.500000

The next table shows how many times you would have to win each match-up to achieve a perfect bracket. For example, the first line shows that in round 1, a one seed has to beat a 16 seed four times. The probability of each game winning is 0.992647, so the probability of winning all four is 0.9926474 = 0.970911.

Round Match-Up Probability
Win
Number
Games
Probability
All Win
1 2 66 35 0.530303
1 3 35 21 0.600000
1 4 63 46 0.730159
1 5 46 38 0.826087
1 6 12 9 0.750000
1 7 7 6 0.857143
1 8 72 58 0.805556
1 9 70 64 0.914286
1 10 6 5 0.833333
1 11 6 3 0.500000
1 12 20 20 1.000000
1 13 4 4 1.000000
1 16 140 139 0.992857
2 3 56 34 0.607143
2 4 7 3 0.428571
2 5 5 0 0.000000
2 6 30 23 0.766667
2 7 81 57 0.703704
2 8 7 2 0.285714
2 9 1 0 0.000000
2 10 50 32 0.640000
2 11 16 14 0.875000
2 12 1 1 1.000000
2 15 140 131 0.935714
3 4 7 4 0.571429
3 5 4 2 0.500000
3 6 73 44 0.602740
3 7 15 9 0.600000
3 8 2 2 1.000000
3 9 2 2 1.000000
3 10 12 8 0.666667
3 11 48 32 0.666667
3 14 140 119 0.850000
3 15 1 1 1.000000
4 5 74 41 0.554054
4 6 4 2 0.500000
4 7 5 2 0.400000
4 8 9 4 0.444444
4 9 3 2 0.666667
4 10 2 2 1.000000
4 12 38 26 0.684211
4 13 140 111 0.792857
5 6 1 1 1.000000
5 8 3 1 0.333333
5 9 3 1 0.333333
5 10 1 1 1.000000
5 12 140 90 0.642857
5 13 17 14 0.823529
6 7 8 5 0.625000
6 8 1 0 0.000000
6 10 7 4 0.571429
6 11 140 88 0.628571
6 14 15 13 0.866667
7 8 2 1 0.500000
7 10 140 85 0.607143
7 11 3 0 0.000000
7 14 1 1 1.000000
7 15 4 3 0.750000
8 9 140 71 0.507143
8 11 1 1 1.000000
8 12 1 0 0.000000
8 13 1 1 1.000000
8 16 1 0 0.000000
9 11 1 0 0.000000
9 13 1 1 1.000000
10 11 3 1 0.333333
10 14 1 1 1.000000
10 15 5 5 1.000000
11 14 6 5 0.833333
12 13 12 9 0.750000

The next table shows the probability of surviving each round, given that you make it to that round. For example, the probability of surviving the 32 games in round one is 1 in 27,156. To get the probability of completing the entire bracket correctly, take the product of the probabilities in the table below. The bottom line is a perfect bracket probability of 0.000000000018, or 1 in 55.4 billion.

### Probability Advancing in Perfect Bracket

Round Probability
Survival
Inverse
Probability
1 0.000036824497 27,155.84
2 0.001284307886 778.63
3 0.038621815874 25.89
4 0.079085421795 12.64
5 0.250000000000 4.00
6 0.500000000000 2.00
Product 0.000000000018 55,380,246,801

By the way, the probability after the 2018 tournament was 1 in 42,743,890,552.

## Warren Buffett Prize

Warren Buffett has offered \$1,000,000 per year, for life, to any employee of his who can fill out a perfect bracket. Warren has about 377,000 employees (). Assuming each one followed the strategy above, and lived 60 more years, the expected cost to Mr. Buffet would be \$408.45.

If you don't want to risk sharing the glory of a perfect bracket with someone else following my strategy, I'd suggest picking one or two 9 seeds to beat an 8 seed. The probability of the nine seed beating an eight seed is 47.8%, so a small sacrifice to increase your chances at hogging the glory of a perfect bracket to yourself.

The following table shows the expected number of wins by seed. This may be useful in analyzing proposition bets like how many games will teams in the Pac 12 conference win. To get an expected number of wins, take the sum of the expected wins for each team, according to it's seed.

### Expected Wins by Seed

Seed Expected Wins
1 3.350000
2 2.364286
3 1.864286
4 1.535714
5 1.114286
6 1.085714
7 0.907143
8 0.721429
9 0.578571
10 0.621429
11 0.600000
12 0.514286
13 0.250000
14 0.164286
15 0.071429
16 0.007143
Total 15.750000

Another common question is, "will a number one seed win?" In the 35 seasons at the time of this writing, a number one seed won 22 times, for a probability of 62.9%.

Another popular one is, "will a 14 to 16 seed win?" Based on the data, I estimate the probability of at least one win of a 14 to 16 seed winning to be 61.1%.

Yet another type of bet will be on the number of number one seeds to make the Final Four. I show the probability of any given number one seed winning it's division, thus making the Final Four, to be 40.7%. Thus, 1.63 number one seeds, on average, will make it to the Final Four. The following table shows the probability of 0 to 4 number one seeds in the Final Four.

### Expected One Seeds in Final Four

Number Probability
4 2.75%
3 16.00%
2 34.96%
1 33.94%
0 12.35%
Total 100.00%

A bet you might see would be over/under 1.5 number one seeds in the Final Four. The probability of under 1.5 is 12.35% + 33.94% = 46.27%.