Combinatorics can involve some messy computation. All the more reason to keep it simple(r) whenever you have a chance.
Barry Bonds has not started 16 of the 91 games that the Giants have played so far this season. There are 260,462,895,672,871,000 (260 quadrillion) possible combinations of games that Bonds might have “chosen” to miss. For example, Bonds missing Giants games # 88, 4, 62, 18, 22, 23, 61, 2, 91, 54, 87, 10, 58, 19, 65 and 83 is one such combination.
Of those combinations, 1,200,635,647,008,340 (1.2 quadrillion) involve each of the three Giants games that have been nationally televised on ESPN this season, as well as any other set of 13 non-ESPN games. In other words, the probability that if Bonds were picking 16 out of 91 games at random to miss, it would so happen that all 3 of the ESPN games were included among those 16, is 1,200,635,647,008,340 divided into 260,462,895,672,871,000, or about 215-to-1 against.
--Nate Silver, Baseball Prospectus Unfiltered
Those numbers seem plausible but I don't think many humans have a good sense of whether hundreds of quadrillions are the right order of magnitude. In a perfect world you could solve this problem without numbers that large. In fact, you can solve the problem int his world without numbers that large.
This problem is equivalent to three friends each drawing (without replacing) a ball out of an urn that contains 91 balls, 16 black and 75 orange. The probability that each of the three friends draws a black ball is:
16/91 * 15/90 * 14/89
(If the first friend draws an orange ball then we know our condition is false. If the first friend draws a black ball then the number of balls and the number of black balls have each decreased by one, etc.)
According to Excel (don't bother with a calculator!) that's 3,360 out of 728,910. Six-digit numbers are still difficult for a human to put in perspective but at least they're not quadrillions.
(And yes, consistent with what Nate said, about 215.9 to 1 against.)
Oh, if you're being truly anal about problem equivalence, you can have 91 people in a line, each of whom will draw a marble without replacing. Three of those people are friends, and they happen to be at places within the line that correspond to the ESPN games on the Giants' schedule-to-date.
Posted by Matt Bruce at July 18, 2007 11:49 AMMy guess is the dude used the quadrillions because that makes it seem like bigger odds. The actual odds of 215 to 1 are given as almost an afterthought and the impression the non-careful reader comes away with is that it's ridiculously unlikely that Bonds missed those three games at random.
Though, from that excerpt, I don't even know if he did miss those three games.
Posted by: mountmccabe at July 20, 2007 09:06 PM