17 March 2008

When Is A Basketball Lead Safe?

*Take the number of points one team is ahead.
*Subtract three.
*Add a half-point if the team that is ahead has the ball, and subtract a half-point if the other team has the ball. (Numbers less than zero become zero.)
*Square that.
*If the result is greater than the number of seconds left in the game, the lead is safe.

From here via NewMexiKen.

Why care? Because this heuristic formula (loosely defined by the author as "a mathematical rule that works even though no licensed mathematician would be caught dead associating with it."), is part of a larger class of partically important problems which the author of the formula cited above (Bill James) describes and NewMexiKen excerpts:

[T]here comes a time when it ain’t over, but … it’s over. There comes a time in a relationship when a woman will still answer your phone calls, but you’re wasting your money buying flowers; you know what I’m saying? There comes a moment during a job interview when you’re still talking, but you might as well take off your shoes. There is a time in an illness when you’re not dead yet, but you might as well stop taking that nasty medicine.

Another example is when a delegate count or vote count becomes insurmountable in a political campaign, even though the results are not yet final, i.e. a point at which a race can be safely called.

Also, while the basketball formula is purely empirical, often formulas that work for one example of a class of problems are similar in form to formulas that work in similar problems.


rtaylor1802 said...

I read your blog as I was filling out my brackets for the NCAA Men's Tournament. I recall a game a few years ago when a team that had been up by nine points with one minute left ended up losing. The team that had been losing kept intentionally fouling the team that had been winning and as the "winning" team could not make a foul shot, the "losing" team made their two-point and three-point shots when they got the ball back.

A miraculous finish?

Not if you use the formula:

9 - 3 = 6 - 0.5 = 5.5
5.5 x 5.5 = 30.25
60 - 30.25 = 29.75

If my math is correct, the team that had been up by nine points only had a little bit better than a fifty-fifty chance of winning the game.

Thanks for your assistance with my efforts to win the office pool!

Rick Taylor

Anonymous said...

Rick, your math is correct, but your interpretation is off. The team up by 9 doesn't have a 50/50 chance of winning, but they are 50% "safe" or half way to having the game locked up. Hope this helps, E.