Fun with Probability and Football
I came across an item on cnn.com sports/sports illustrated: "Predict the exact NFL standings and win $1 million." At first this sounded pretty hard, but maybe not impossible. But then I started to do a little math.
Suppose we knew nothing about football and chose our standings at random. How many possible combinations are there? Each division has 4 teams, so the standings in each division can turn out 4*3*2*1=24 different ways. There are eight divisions. Therefore there are 24 raised to the 8th power possible ways the standings can turn out. That's 110,075,314,176 ways, which is more than even a math nerd like me would have guessed. 110 billion. There are only about 80 million combinations in the Powerball lottery, so the likelihood of randomly guessing the NFL standings is less than 1/1000th that of winning Powerball.
Of course our guesses wouldn't be completely random: we know a little football. Quantifying that is pretty difficult, though. Suppose for each division we knew for certain where one team would finish. This is unrealistic, obviously, because in reality we don't have 100% information on one team and none on the rest -- we have a little information on them all. Still, it seems to me to be approximately the right amount of information overall. If we're spotted that one team per division's finishing order, the number of combinations for the rest drops way down to 1,679,616. If you win $1 million for every 1,679,616 times you play, that's about 60 cents per play. If it takes you 5 minutes to fill out your picks, you're making $7.20 per hour, which is about what they pay at our nation's largest employer, Walmart, and it comes with a similar healthcare plan.
Unfortunately I clicked the link to the contest and discovered some fine print. Not only do you have to predict the standings order, but also which teams go to the playoffs and the playoff results through the Superbowl. The odds of doing that are surely many thousands to one against, so it looks like you're better off at Walmart after all.
