Billionaire Warren Buffett is giving away $1 billion (references 1, 2, 3) to anyone who can pick a perfect March Madness bracket.
Roughly speaking, what is the probability that Buffett will have to pay up?
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Depends how many entries are made. – André Nicolas Jan 28 '14 at 00:43
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I think this is duplicate. – Potato Jan 28 '14 at 00:43
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Possible duplicate: http://math.stackexchange.com/questions/648321/what-are-the-exact-odds-of-getting-a-perfect-ncaa-bracket – Baby Dragon Jan 28 '14 at 01:06
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1It's not quite a duplicate; other questions ask for the probability that one person wins. This asks for the probability that any person wins ("what is the probability that Buffett will have to pay up?"). – John Feminella Jan 28 '14 at 12:40
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Great question! Think about this: Buffett can afford to lose the money, but all the same, he probably wouldn't be offering it if he didn't think he'd come out ahead. So it's safe to assume that the probability will be small.
This is the Mathematics Stack Exchange, though, so let's see some math.
First, the very best handicappers can manage to pull off a win rate against Vegas odds of around $p = 55\%$, which is only slightly better than random guessing.
March Madness is a 64-game or 68-game single-elimination tournament depending on whether we're talking about men's basketball (68 teams) or women's basketball (64 teams). Let's pick 64 teams just to give you a slight edge. If there are 64 participating teams and one eventual winner, then there are $n = 63$ matches, since each match reduces the number of remaining teams by $1$.
The probability of winning $n$ coin flips in a row, where you have a probability of $p$ to win each flip, is $p^n$.
In this case, that's $P(\text{individual guess is correct}) = 0.55^{63} \approx 4.34 \cdot 10^{-17}$ for an individual person to correctly guess all 63 games, if that person was as good as the best handicappers.
To put that in perspective, it's roughly 1 billion times easier to win the Mega Millions jackpot than it is to win Warren Buffett's bet.
But that's a little different than the question, which asks for the probability that someone will be able to guess correctly. If one person's probability is $P$, what's the probability if everyone gets a shot?
That's a Bernoulli trial with at least 1 success in $\sim 7$ billion people, and the odds don't improve much -- it's about $4.61 \cdot 10^{-14}$, or roughly 1 million times easier than winning the Mega Millions jackpot.
So, don't quit your day job.
John Feminella
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2@AsafKaragila I do not know the particulars of your financial situation, so I cannot give you good advise. – Baby Dragon Jan 28 '14 at 01:26
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@BabyDragon: Well, for one, my night job is staring that MSE and answering axiom of choice questions. I'm looking forward for the day where my vast MSE reputation will be translated to quality whisky. – Asaf Karagila Jan 28 '14 at 01:29
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@AsafKaragila We could attempt to set up a futures market based upon MSE reputation, then manipulate the market. – Baby Dragon Jan 28 '14 at 01:30
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@JohnFeminella is just trying to throw you off so that if UVa wins, he gets all the winnings. – Vladhagen Mar 19 '14 at 21:47