March 17, 2011
Perfect NCAA Bracket Odds (realistic)

Hoping for a perfect bracket this year?

With contests online giving away 159 even 100 million dollars on a perfect bracket, the odds MUST be slim, right?

Right.

I’m a proud math nerd, and my favorite aspect of math is probability. Let’s get started:

Assumptions:

  • You’re filling out a 64 team NCAA bracket (ignoring play-in games). FYI that means there are 63 games. In any single elimination tournament there is always 1 game less than the number of teams—since each game eliminates one team and all but one team (the winner) must be eliminated.
  • You’re a better than average handicapper:
    • For four games (1 vs 16) you have a 99% chance of being correct
    • For another four games (2 vs 15) you have a 90% chance of being correct
    • For 8 games you have a 75% chance
    • For all remaining games, you’re so good, you can pick it right 60% of the time.

My assumptions are fairly ridiculous in that no handicapper is that good, but what the heck. It helps illustrate my point.

Your odds of having a perfect bracket are:

(0.99^4 * 0.90^4 * 0.75^8 * 0.60^47)

1 in 423,535,978,540

That is 1 in 423 BILLION. Remember, this assumes that you’re one heck of a handicapper. Change your odds to just 55% for the remaining games and it goes down to:

(0.99^4 * 0.90^4 * 0.75^8 * 0.55^47)

1 in 25,290,223,679,994

That is 1 in 25 TRILLION.

OK, let’s not get out of control. Let’s stick with the 60% number of 1 in 423 billion. Let me help you understand how impossible that is to achieve:

Here is your 1 in 423 billion challenge:

Find the right ping pong ball.  That’s it! (well, sorta…)

Somewhere there is a secret ping pong ball. Say its marked with an invisible ultraviolet marker.

You’re standing in the middle of the Indianapolis Colts football field. You are chest deep in a sea of  ping pong balls (5 feet deep), covering the entire playing field.  You need to pick the ONE ping pong ball that is secretly marked.

As you wade around you feel balls breaking under your feet. You wonder if that was the one. You move and a wave of balls moves with you, both away from you and filling in behind you. Which way to go? How deep to grab?

As you ponder the choices, you’re told that the ball could be at any NFL stadium in the country, or any college football stadium in the country, division I, II or III. There are also 50 balls on every seat in every stadium and 6,000 cars at each stadium—each car is also filled with ping pong balls. Pick your stadium, choose the parking lot, the field or the stands, and try to pick the one special ball, that looks like all the rest.

That’s 1 in 423 billion. And remember, you have to be a great handicapper to even get those odds.

Good luck!

————————

(for you fellow math nerds, here’s how I calculated it)

A ping pong ball is 1.5748 inches wide, has a volume of apprx 2.05 cubic inches. (sphere volume = 4/3 * pie * radius^3)
So by volume, 845 ping pong balls could fit in a cubic foot (12^3 / 2.05). But there will be space between balls, since they are not cubes.
Maximum packing density for spheres is 74%, so about 625 balls could actually fit in a cubic foot (845 * 0.74).
A football field is 360 feet (100yds + two 10yd endzones) long and 53yds wide (159 ft), so it has 57,240 sq feet.
So on average, each 1 foot deep in ping pong balls the field is, there are 57240 * 625 balls. That’s 35,775,000 balls per foot deep.
There are 32 NFL stadiums, 244 division I stadiums, 157 D2 and 238 D3 stadiums. I generously averaged 30,000 seats per stadium, which is surely way too high (the real number is probably half that, so there could be twice as many balls per seat). The average large car has 120 cubic feet in it. So each car holds about 75,000 balls. All in all, each stadium has about 629,235,000 balls. With 671 stadiums total, we have about 423 billion balls to choose from.

stadiums (671) * [square feet on a field (57240) * balls per cubic foot (625) * depth of balls in feet (5) + average number of seats (30,000) * balls per seat (50) + number of cars at each stadium (6000) * balls per car (75000)]= apprx 423 billion