No. of Recommendations: 6
https://frontofficesports.com/warren-buffett-march...For a decade, Warren Buffett, the billionaire "Oracle of Omaha" and Creighton basketball fan, has held a men’s NCAA tournament contest for employees of his Berkshire Hathaway conglomerate. Berkshire has around 395,000 employees in total among all its 60+ subsidiaries, and the contest usually gets between 60,000 and 70,000 entrants.
The full prize: $1 million. But no one has ever won the jackpot—until now.
In the past, the prize was $1 million every year for life for anyone who perfectly predicts the Sweet 16; in the 10 years of the pool, it hasn’t happened. The consolation prize was $100,000 to the bracket that remained perfect the longest; someone won that prize every year, or multiple people split it.
This year, Buffett tweaked the rules, effectively lowering the bar to earn the million bucks: He said he’ll give $1 million to anyone who correctly nails at least 30 of the first 32 games. Buffett told the Wall Street Journal he eased the rules because "I'm getting older ... I want to give away a million dollars to somebody while I’m still around as chairman." Buffett is 94.The math's not so hard. You ignore the bracket structure and with 64 teams, there are 63 total games, and 2^63 ~= 9,200,000,000,000,000,000 possible brackets. If you have even odds, your odds of getting it right are 1/(2^63), or (1/2)^63, or ~1.1 * 10^-19
If you're better predicting basketball than me and get 60% correct instead of 50%, it's (6/10)^63, it's ~1.1 * 10^-14. The 10% edge in odds per game makes it ~100,000 times more likely you'll guess them all correctly.
These are special cases of the binomial distribution. Named that way for the binomial binomial coefficients, which are the integer coefficients of (x+y)^n. If you multiple that equation out, you get all possible permutations of n x's and y's. But for ... cases most are familiar with, the multiplication of numbers is commutative, so (x+y)^2 = xx + xy + yx + yy = x^2 + 2xy + y^2, and the second coefficient is 2 because xy = yx. In combinatorics, you'd call the coefficient of the kth term out of n terms "n choose k" or C(n,k) and is equal to n!/(k!*(n-k)!). In the case the 2nd degree polynomial earlier, the middle term with coefficient equal to 2 could also be computed by 2!/(1!*(2 -1)! = (2*1)/(1*1) = 2.
Back to the first equation, to get the probability of those outcomes, you can use the probability function of the binomial distribution, P(k,n,p) = C(n,k) * p^k * (1-p)^(n-k). In English, "the number of different ways you can be right k times and wrong the rest of the time, times the odds of being right k times, times the odds of being wrong the rest of the time". So getting 63 out of 63 right in your bracket can now be computer as C(63,63) * (1/2)^63 * (1/2)^0 = 1 * (1/2)^63 * 1 = (1/2)^63. A slightly harder way to get the same answer.
So to Buffett's new challenge, the odds of getting all 32 of the first round right if you're flipping a fair coin are (1/2)^32, or 2.3*10^-10. To get 30 or more right, you just need to add them up the odds of getting 30, 31, and 32 right: P(30, 32, 1/2) + P(31, 32, 1/2) + P(32, 32, 1/2). Or,
(C(32,30)* (1/2)^30 * (1/2)^2) + (C(32,31) * (1/2)^31 * (1/2)^1) + (C(32,32) * (1/2)^32 * (1/2)^0) =
((1/2)^32 * (32*31/2) + ((1/2)^32 * 32) + ((1/2)^32 * 1) =
(1/2)^32 * (496 + 32 + 1) =
529 / (2^32) ~=
1.2 * 10^-7 or about 1 in a million.
Since Berkshire employees include a bunch of actuaries, lets give them the same 60% odds of guessing each game's outcome correctly
(C(32,30)* (6/10)^30 * (4/10)^2) + (C(32,31) * (6/10)^31 * (4/10)^1) + (C(32,32) * (6/10)^32 * (4/10)^0) =
496 * (.6)^30 * (.4)^2 + 32 * (.6)^31 * (.4) + (.6)^32 =
0.0000175 + 0.0000016 + 0.0000001 ~= 0.00002, or 1 in a fifty thousand.
My bet, with 400,000 employees, assuming they knew ahead of time about the new rules, is 2 or 3 winners being announced tomorrow. Without many upsets, I wouldn't be so surprised if it were higher.
Sorry for any math errors, particularly to Manlobbi. I am rusty. I love that in this forum, I suspect, this is old hat for most commenters.
