## Thursday, January 23, 2014

### The Billion Dollar Bracket

Yesterday it was announced that Quicken Loans would pay \$1 Billion Dollars to someone that picked a perfect NCAA bracket.  It wouldn't be Quicken Loans that would actually shell out the cash, it would be Warren Buffet's Berkshire Hathaway that would pay it.  The number of entrants is limited to 10,000,000 people though.  It seems to be a little early to think about brackets but it brings out kind of a fun exercise.
When I'm not taking pictures, I'm a quality engineer which means that I get to play around with statistics.  A subset of statistics is probability.  And this presents a pretty interesting probability problem.  If someone is willing to shell out a billion dollars, it must be pretty long odds for someone to pick a perfect bracket.  Just how long do you ask.

Well, let's take the easy exercise first.  In the first round, you have 32 games with one of two probably outcomes.  The second round has 16 games.  The third has 8 and so on.  In total there are 63 games.  If we assume a 50/50 outcome, the probability of picking a perfect bracket is 2^63 or roughly 10^19 power.  This means that there are 19 zeroes after the 1.  Which means using straight up odds, the probability of picking a perfect bracket is 1 in 10 Quintillion.

However, we know that it is not an equal probability of outcomes.  For instance, a number 16 seed has never beat a 1 seed.  The 2 seed wins roughly 94% of the time.  The 3 seed wins 85% of the time.  The 4 seed will triumph 78% of the time.  The 5-12 matchup is famous for the annual upset.  The reason being is that the 5 seed is typically a team from a major conference on the way down and the 12 seed is typically a mid-major (or smaller) conference champion.  Even with that scenario, the 5 seed wins 65% of the time.  The 6 seed will emerge victorious 66% of the time.  The 7 seed emerges 60% of the time.  The 8-9 matchup is the closest we have to a 50/50 shot, but the 8 seed emerges 48% of the time.  Granted these numbers are rounded a bit for ease of calculation.  These all come from data since the 64 team tournament emerged in 1985.

Now for the probability of picking a perfect 1st round.  We can ignore the 1 seed games because these are pretty certain (at least based on current data).  There four games for the other rounds, so our formula becomes (0.94^4)*(0.85^4)*(0.78^4)*(0.65^4)*(0.66^4)*(0.60^4)*(0.48^4) or 3.5 X 10^-5.  To make it more like the above numbers we get roughly a 1 in 28,000 chance to pick a perfect first round.  Now I know that we can't assume 50/50 matchups in the rest of the tournament, but I don't feel like looking up those numbers so I will just use those probabilities.  So for that we 6 X 10^13 or about 1 in 60 Trillion.  This number would probably represent the chance that someone who lives and breathes college basketball would pick a perfect bracket.  Limiting it to 10,000,000 entrants means that Warren Buffet has roughly a 1 in 6 X 10^8 chance of even paying out.  So I think his billion dollars is safe.

Still, it would be pretty cool to think what I could do with a billion dollars (well 667 million after Uncle Sam gets his take).