For The Love Of Karl Friedrich Gauss, Foul Him! Print E-mail
Written by Constantine Scionti   
Wednesday, 19 December 2007
A gripe that has gotten me to shout obscenities at the TV for years. Okay, so you're up by three points with a few seconds left in regulation, you have no fouls to give, and the opponents are inbounding the ball.  Do you foul and send someone to the free throw line for two shots, or just play straight-up defense and make the three-point attempt as hard as you can?  The standard answer, as evidenced both by what teams do, and by what commentators recommend (most of them old coaches and/or players) is not to foul.  Here's why they are wrong: Bear with me, there is going to be some math involved, but nothing too scary.  Bob Cousy has often pointed out when doing analysis for the Celtics that basketball is a game of percentages.  Wouldn't want to disagree with the Cooz, so let's see what that means in this case. The best three point shooters in the league are over 40% from beyond the arc, some of them well over.  Last season the leader in three-point percentage was Jason Kapono, at 51.4%.  In fact, all of the top twenty were over 40%.  But that's not under pressure at the end of the game, the old-schoolers will respond.  True, but do you really want the ball finding its way to, say, Robert Horry for a potential game-tying three-pointer at the end of regulation?  Let's call it a 35% shot, if the ball gets to an open shooter, which can usually be accomplished with a pass or two.  So that's 35% that the game ends up tied and you end up in overtime.

Now, let's examine what happens if you foul deliberately and send someone to the free throw line.  An excellent free throw shooter hovers around 90%, so that's the number we'll use.
A brief excursion into mathematics (sorry): to get the odds of two things happening, you multiply the percentage of each event.  So, if you want to know the odds of two coins both landing heads, that's .5 x .5 = .25, if you want the odds of three coins all landing heads, that's .5 x .5 x.5 = .125.  

This is relevant for this case because a series of events has to happen for a team to tie the score if someone is shooting two free throws when down by three:  The shooter has to make the first free throw, then miss the second free throw (but hit the rim, or it's a violation, and the ball goes to the other team); then the team needs to get an offensive rebound off the missed free throw, and hit the game-tying shot.

So, what are the odds of all of those things happening?  This is conjecture, of course, but here are some numbers I crunched:
Our excellent free throw shooter has a probability of .9 of hitting the first shot.  I have no idea what the probability of hitting the rim hard enough to have a chance at an offensive rebound is, but this guy's a great shooter, so we'll give him 95% (.95 in the probability world).  So far, that's .9 x .95, or .855.

What's the chance of getting an offensive rebound when everybody in the building knows that is what you are trying to do?  Well, the defense surely put their best rebounders in, so let's give it 40% (and I strongly doubt it's that high).  Now we're down to 34.2% (.855 x .4).

So now the offensive rebounder has gotten the ball and controlled it.  Well, your rebounders are pretty tall, and they are still between the offensive rebounder and the basket with their hands up, right?  Let's make it really nightmarish and say that the offensive rebounder who just got the ball is Tim Duncan, who shot 54.6% last year.  Most of the shots he took during the season were not under the defensive pressure that he would be facing in the lane with a second or two left in the game, but he's great under pressure.  So great, I'll say he'll have a 55% chance of making this shot.  Well, that results in .342 x .55, or 18.8%.  Oh, and this all has to happen in a couple of seconds--no time for pump fakes or low post moves.

So, with my numbers (generous ones, at that), we have an approximately 19% chance of fouling resulting in a tie game and overtime, while letting a good shooter get off a three-pointer has a much higher chance of success.

Why does nobody do this?  I can think of two reasons.  First: they have not gone through the math to see which is more likely, a made three-pointer, or the series of events described above.  

Second: fouling is the more remarkable choice.  Even though it's mathematically better, it will still fail some of the time, and when it does, the next day all the sportswriters and talk-show hosts will scream about what an idiot the coach is.  It won't matter that it fails approximately half as often (35% to 19%), if two teams each face this situation ten times in a given season and one never fouls and the other always fouls, the team that allows the shot will end up in overtime approximately four times and no one will say anything about it--hey, you play straight-up defense and sometimes they hit the shot anyway, what're ya gonna do, right?  The team that fouls will end up in overtime approximately twice, and if the coach does not have the job security of Gregg Popovich or Phil Jackson, there will be calls for his head on a stake.
  If you're already under the gun as a head coach, you probably don't want the headache of explaining probabilities during a press conference in addition to all your other headaches.
[By the way, for non-math geeks: Gauss was a great mathematician of the early 19th Century; the bell curve is also called Gaussian distribution.]

 

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