Down by 3, under a minute left

Orlando was faced, yesterday, with a common occurrence in basketball games. Down by 3, with the ball, and on its last possession (not counting fouls, turnovers, etc). The dominant strategy in the NBA has always been to go for 2, foul, and hope that the other team misses 1 or more free-throws, and quickly make another 2.

While this might be the risk-averse strategy, attempting to extend the game, it is not likely the correct decision - or, at least visits a revisit.

Ok, so, we'll have to start with some assumptions: lets assume that your probability of making a 2 point basket, when down by 3 is pretty high (since the other team will not likely foul and put you on the line with a chance to tie the game) - lets put this probability at 60%. Lets assume that the free throw % for the opposing team is just 70% (low, if you end up putting Rip Hamilton on the line). Lets put the probability at making your 2nd 2 point shot at 50% (here the opposing team is playing full out defense to try to prevent you from winning the game, and the refs are notorious for not calling fouls in this situation). If these are true, your probability of tying the game is: .6*(.3+.3-(.3*.3))*.5 = .15, or just 15%. Your probability of winning the game (ie - the opposing team misses both free throws), in this situation is just 2.7%. If the free throw % of the other team rises, your chances at tying or winning become even more minuscule.

(note on assumptions: last night, Orlando shot 55% from 2, and 36% from 3. Detroit shot 87% from the line. Jameer Nelson was 41% from 3 on the season, and was 2/3 last night). Using these figures, the probability of tying the game is 8%, and the probability of going up by 1 is just .6%

This means, that if you can hit a 3 pointer at just 15% (or even as low as 8%), you have a better shot at tying the game with a 3 pointer. Even if the other team (Detroit, in this example), gets another shot at the basket, you still have a much better chance at sending the game into overtime or winning the game if you shoot the 3-ball. With Jameer Nelson averaging over 41% for the season (and shooting well last night), he ties the game with probability 41%, and Detroit then wins the game with a buzzer beater with probability 36% (Detroit's field goal % last night), putting the possibility of the game going into overtime at 27%, as opposed to 8%. If its under 24 seconds, as opposed to just under a minute, the calculations would much more favor shooting a 3, rather than a 2, as you can hold until last possession, and tie the game with the probability of tying and sending the game into over time at 36% (Orlando's 3 pt % last night), as opposed to 8%.

Ok, so, there are some simplifications and weaknesses here. This ignores any dumb turnovers by Detroit, but, since they only had 3 turnovers on the entire night, the probability of them not being able to inbound the ball seems slim. This ignores offensive rebounds and second chance points, but - 3 point shots create opportunities for long rebounds and more second chance points. Obviously all the assumptions made determine the outcome, but, I use assumptions that highly favor shooting a 2 pointer over a 3 pointer.

The conclusion of this is that it seems that under any reasonable specification, the dominant strategy ought to be to take the 3, rather than get 2, foul, and make another 2.