March 29, 2011

Score tied: should you foul?

Filed under: Uncategorized — wwinston @ 10:27 pm

Henry Abbott of the great site has created quite a stir by suggesting that if  the score is tied and your NBA opponent has the ball with <24 seconds to go in an NBA game you should foul them. I did a simple analysis of this situation and the decision is not clear cut.

    usfg 0.4          
    themfg 0.4          
    usotwin 0.5          
    themft 0.7          
    do not foul foulwin        
    0.3   0.329        
          foul chance I win  
          0made 0.09 0.7  
          1 made 0.42 0.4  
          2made 0.49 0.2  
      FT chance
0.029 0.6 0.65 0.7 0.75 0.8 0.85
0.3 -0.048 -0.0705 -0.092 -0.1125 -0.132 -0.1505
FG chance 0.35 0.014 -0.00912 -0.0315 -0.05313 -0.074 -0.09413
0.4 0.076 0.05225 0.029 0.00625 -0.016 -0.03775
0.45 0.138 0.113625 0.0895 0.065625 0.042 0.018625

In the example shown I assumed each team hits 40% of their endgame field goals and the team with the ball makes 70% of FT’s and is in the bonus. I also assumed that each team has a 50% chance of winning in overtime. I assumed nobody takes any 3 point shots.  For this situation if we do not foul we win only if opponent misses and we win in OT. This happens with probability .5 (1-.4)=.3. If we foul opponent makes – FT’s 9% of time, 1 FT  42% of time and 2 FT’s  49% of time. As Byron pointed out, I am assuming fouling team rebounds all missed FT’s.

  • If opponent makes 0 FT’s we win if we make our last shot or win in OT. This occurs with probability .4 +.6*.5=.7.
  • If opponent makes 1 FT we win if we make our shot. This occurs with probability .4
  • If opponent makes 2 FT’s we win if we make our shot and win in overtime. This happens with probability .4(.5)=.2

Therefore if we foul we win with chance .09(.7)+.42(.4)+(.49).2 = .329.

Thus in this situation you should foul!!

The table above gives our chance of winning-opponent chance for various scenarios. The red numbers correspond to situations where fouling is beneficial.

Of course in a college game where other team is in 1-1 fouling is even better!

I also assumed that the opponent is equally likely to shoot any value between 65% and 85% from the foul line and each team is equally likely to  shoot any value between 30% and 45% chance of making a FG. Then fouling gives you a 2% smaller chance of winning than not fouling.


  1. Interesting…

    But I think there also needs to be some adjustment for if the fouled team gets the rebound and therefore retains possession.

    Obviously, this would happen on merely the last free throw…so say a 30 percent chance of a miss, and just guessing here that the shooting team might get the rebound 25 percent of the time in such a high-pressure situation. Of course, if they did…they would likely be fouled again.

    That would change the expected outcome.

    Comment by Byron — March 29, 2011 @ 10:49 pm

  2. Isn’t a shot designed to occur at the end of the game a lower percentage shot than the average shot? Notmall, when you take a shot, you have the entire 24 second clock to work with and the opponent doesn’t know when you will shoot it an has to adjust their D accordingly. However, if you’re going for the last shot, say in the last 5 seconds, the plays you can run are more limited and the opponent has a better idea of what is coming. 82 games has the percentage of made shots with 24 seconds or less left in the game, team with the ball is either tied or down by 1 to 2 points, at 29.2%. Granted this includes guarded threes and full court heaves, but it shows that fouling is a really bad idea unless a very poor FT shooter, like Shaq or Dwight Howard is the one with the ball. At the end of the game, that is a very rare sight unless it’s after a rebound or leads to a wide open shot.

    Comment by Mo — March 31, 2011 @ 2:42 pm

  3. @Mo…that 29% is better not just because of the shots you mention. There’s also the possibility of you being fouled on the ensuing possession. For example, Billups also got 12 free throws, which I’m sure would raise his clutch TS% since he’s an excellent FT shooter.

    I think something like this would be better on a case by case basis, where a team knows certain plays/player succeed at a certain rate, or if like you said a particularly bad foul shooter can be fouled. Also, if a team is up against an inferior opponent that they have more than 50% confidence they can beat in OT. As a sweeping observation, the advantage is so minimal that it’s not worth advocating as a general strategy.

    Comment by luislandry — March 31, 2011 @ 3:50 pm

  4. I don’t think Abbott made the argument you are attributing to him. His argument was that there may be situations near the end of periods where it is worth it to foul to get an extra possession. Timing it right, you could get two or three extra possessions per game that way, so the issue is what is that worth. Fouling at the end of the game with a particular scoreline is a subset of the general proposition, but analyzing that doesn’t answer the question of whether, in the long run, three extra possessions a game is worth giving up six free throw attempts per game.

    Comment by John Jenkins — March 31, 2011 @ 7:51 pm

  5. What about turnovers? It seems like the average team commits 14 turnovers per game and has an average of 96 possessions which gives us about a 15% chance of turnover with no foul.

    Comment by Naasir — March 31, 2011 @ 8:13 pm

  6. he discussed it on NBA today podcast last friday

    Comment by wwinston — March 31, 2011 @ 10:04 pm

  7. i do not think teams turn it over much when playing for last shot but i should put that in.

    Comment by wwinston — March 31, 2011 @ 10:06 pm

  8. Wayne – a huge factor is that possessions aren’t worth 1.07 points anymore at that point. You’ve estimated 40% FG without 3s. I compiled 82games data and got around 45% eFG% (including 3s):

    We also have to think beyond turnovers. The thing that jumps out to me about the “no-foul” probabilities is that you assume the defending team can ONLY win in OT. Obviously, they have a small chance to win in regulation with their own possession when the offense leaves a little time on the clock (make or miss) or turns it over. I don’t think that’s an insignificant probability to discard…

    Comment by ElGee — April 1, 2011 @ 3:53 pm

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