Henry Abbott of the great Truehoop.com 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.
|do not foul||foulwin|
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.