I'm not too surprised that Warren Buffet could stop feeling nervous about having insured Quicken Loans' billion-dollar payout for a perfect NCAA basketball bracket before the first round of the tournament was completely over. The chance of a perfect bracket has been widely reported to be 1-in-9.2 quintillion. The source of that incomprehensible number is 2^{63}, the exponent representing the 63 games in the tournament and the base representing the two possible winners of each of these games. Since there is obviously not an equal chance of either team winning any particular game, I think this number is too high. In the first round of the tournament, for example, a #16 seed has never defeated a #1 seed, so if you pick the #1 seed to win their first game in each of the four regions, and an unprecedented upset doesn't occur, the number of all possible remaining outcomes shrinks to 2^{59}.

But Buffet really wasn't taking any chance at all. The chance of someone winning was more theoretical than actual. So it was just free publicity. Plus, Quicken Loans, which got a lot of personal data about all the contest entrants, paid Buffet an undisclosed fee.

It's fun to try and gauge in a rough sort of way why Buffet wasn't actually taking a rash gamble. Let's consider the difficulty of predicting correctly the outcome of every game in the *first round* of the tournament. There are 32 first-round games, so if you just guessed, the chance of getting every game right is given by 1/2^{32}. This denominator is 4,294,967,296. At least it's small enough so that we have recognizable words for it: just under 4.3 billion. I think I read somewhere that only one entry per household was permitted. The population of the country is a little more than 300,000,000, and I'm pretty sure there are considerably fewer than half that many households. But let's just observe that 150,000,000/4,294,967,296 is about 0.035. So now let's assume that everyone wants Warren Buffet to have to pay out and that all hundred and fifty million competitors collude to cover as many different outcomes as possible--no duplicate brackets, in other words. They still cannot cover even 4% of the possible outcomes. If there was collusion among the maximum number of permitted entrants, how many of the 96-plus percent of uncovered outcomes would have included Mercer over Duke?

All of them, or for sure almost all. And, remember, we are talking about the difficulty of submitting a bracket that's perfect for the first round only.

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