A Quick 3 or a Quick 2?

Nearly every day in the NBA (playoffs included), there are close games that come down to the wire. We see teams with 3, 4, or 5-point deficits with only a shot-clock remaining quite often, and one of the questions commentators always ask during this situation is:

Do you go for the quick 2 and intentionally foul (and hope that the opponent will miss a free throw) or go for a three?

A lot of the time (we saw this with the Houston vs Golden State series), teams decide to go for the quick two, but other times, they go for the three (in the Golden State vs Portland series). Although it was a 3-point game in the latter game, versus a 5-point deficit in the second one, there should be some simple analytical way of determining when a team should go for the 3 or go for a 2.

Golden State is known for its shooting. Although we mainly consider Golden State to be a great 3-point shooter, GSW is full of great free-throw shooters as well. In the clutch, Steph shoots 97 percent from the line, which is one of the best in the league. In addition, Klay shot 100 percent from the line in the clutch this season, while Durant shot 91 percent. This means that the Warriors have at least 3 great options (out of 5) to give it to when they have a small lead with little time left.

Given all of these options, we’ll assume that the free throw percentage for the Warriors when they are intentionally fouled is 90% (which is obviously conservative). 
In addition, we will assume that the average 2 point percentage in the clutch is 55 percent (for the opposing team)while the average 3 point percentage in the clutch is 30 percent (for the opposing team).

Now assuming it is a 3 point game and the opposing team has the ball, we have a couple scenarios:

1. Shoot a 3 -> tie the game
2. Shoot a 2 -> intentionally foul -> opposing team misses a free throw -> Shoot a 2
3. Shoot a 2 -> intentionally foul -> opposing team misses both free throws -> Shoot a 2
4. Shoot a 2 -> intentionally foul -> opposing team makes both free throws -> Shoot a 3

In all of these cases, the opposing team will catch up, but is it more likely to beat the Warriors by shooting a 3 on that play or shooting a 2 on that play?

Well, when you compute the probabilities using some simple multiplication, you get that the probability of winning when you shoot:

A THREE = 31.52%
A TWO = 19.11 %

Clearly, you should go for the three, and no matter what Mark Jackson and Stan Van Gundy said, this is the case.

However, this is only for the Warriors case. What if you have a bad free throw shooting team. Well, we can represent this as a graph, where the independent variable is the probability that the team with the lead makes their free-throw and the dependent variable is the probability of tying the game. (we are going to assume that the 3PT%=30%, while the 2PT%=55% for all teams)

In the above graph, the green line represents taking a three while the white line represents taking a two. Clearly, teams should always shoot the 3 when they are down 3 and have the ball.

Now let’s see what happens if it’s a 4 point game:
There are more possibilities if its a 4 point game, but we’ll remove some of them when we are generating the model (the ones that are essentially negligible because there is such a low probability that it will occur).

1. Shoot a 3 -> intentionally foul -> opposing team misses a free throw -> Shoot a 2
2. Shoot a 3 -> intentionally foul -> opposing team misses both free throws -> Shoot a 2
3. Shoot a 3 -> intentionally foul -> opposing team makes both free throws -> Shoot a 3
4. Shoot a 2 -> intentionally foul -> opposing team misses a free throw -> Shoot a 3
5. Shoot a 2 -> intentionally foul -> opposing team misses both free throws -> Shoot a 2
6. Shoot a 2 -> intentionally foul -> opposing team makes both free throws -> Shoot a 3 ……

When you are playing against the Warriors, the probabilities are like so:

A THREE = 10.43%
A TWO = 5.47 %
Again, we see that the probability to tie is higher when you go for a three versus if you go for a two. Again, we represent this with a graph, with the same X and Y variables.

Interestingly, it is better to go for the two when the opposing team has bad free throw shooting (< 79%) and you are down 4. However, if you have a high FT%, you should always go for the 3. At ~79% FT% it does not matter whether you go for the 2 or if you go for the 3.

Finally, we’ll look at the 5 point game case (the hopeless cause, basically).
At this point, you can basically accept the loss if you are playing against the Warriors. You need basically everything to line up in your favor (missed free throws, made threes, made twos). More concretely you need one of the following:

1. Shoot a 3 -> intentionally foul -> opposing team misses a free throw -> Shoot a 3
2. Shoot a 3 -> intentionally foul -> opposing team misses both free throws -> Shoot a 2
3. Shoot a 3 -> intentionally foul -> opposing team makes both free throws -> Take the L, basically
4. Shoot a 2 -> intentionally foul -> opposing team misses a free throw -> Take the L, basically
5. Shoot a 2 -> intentionally foul -> opposing team misses both free throws -> Shoot a 3
6. Shoot a 2 -> intentionally foul -> opposing team makes both free throws -> Take the L, basically ……

Against the Warriors:

A THREE = 1.59%
A TWO = 0.647%
With that probability to tie up the game, you should just take the loss 😞. Below, is the same graph as above, but for a five-point game.

CONCLUSION: There are different situations when teams should go for the 3 and the 2, but when it’s a 3 point game, ALWAYS go for the 3. Also, when you play the Warriors, you should always go for the 3. We should expect at least 2 close games this year during the finals, with the Raptors on the losing side. If the Raptors can play by this strategy, they should be able to win at least one of these games.

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