### Basketball study: should you foul in the dying seconds when three points ahead?

Your basketball team is up by three points with only a few seconds left to play. The other team has the ball. Which defensive strategy do you choose?

Strategy 1: play defense and try to prevent the other team from sinking a three point shot.

Strategy 2: immediately foul the other team, giving them two foul shots. They will try to sink the first foul shot. But they can’t sink the second, because then they will lose possession while still down one point, thus losing the game. Therefore, they will deliberately miss the second shot, hoping to get the offensive rebound and tip in a two point shot to tie the game. (If they missed the first foul shot, they will try for a desperation three point shot after the deliberate miss.)

In his study "Optimal End-Game Strategy in Basketball," author David H. Annis compares the chances of winning with each of the two strategies. The method is pretty simple: Annis creates two “decision trees,” which are flow charts of how the play will go depending on what happens, and calculates the probabilities from the charts. It’s a bit boring reading while he goes through the theoretical formulas, but picks up once he assigns actual probabilities. For his example, he uses:

Probability of making a free throw = 0.75;

Probability of getting an offensive rebound after deliberately missing a free throw = 0.15;

Probability of a two-point tip-in after getting the offensive rebound = 0.7;

Probability of successfully sinking a desperation three-pointer after an offensive rebound with almost zero time left on the clock = 0.1;

Probability of successfully sinking a non-desperation three-pointer if not fouled = 0.25;

Probability that the offense will be unable to even attempt a non-desperation three point shot = 0.1;

Probability that the defense will get the rebound after a missed three-point shot = 0.7.

Under these assumptions, the probability of winning if you foul the other team is .9588. The probability of winning if you don’t foul them is much less -- .8661. So you should foul.

The author concludes, “for virtually any reasonable values of these probabilities,” fouling is the better strategy.

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Addendum: In the comments, "alan r." points to a nice study by Adrian Lawhorn at 81games.com. Lawhorn examines the same question, does a similar calculation, and comes to conclusions that are roughly equivalent.

One difference is that Lawhorn's non-foul probability works out to 0.8 instead of 0.8661. I think that's partly because he doesn't consider the possibility that the offense can't get a shot off (which Annis estimated at 10%), and partly because (to his credit) he used actual game data from that situation.

## 2 Comments:

Adrian Lawhorn also analyzed the foul/no foul question:

http://www.82games.com/lawhorn.htm

Thanks! I've added a note about Lawhorn's study.

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