Decisions at the end of basketball games often decide the outcome. There are many difficult choices to be made with shot selection, pace, defensive style, and more. However, basketball teams and coaches may not be maximizing their win probability based on their choices. In this article, I will look at the win probabilities of 3 late game scenarios: fouling or defending when up by 3, shooting a 2 to tie or a 3 to win, and shooting a quick 2 or shooting a 3 to tie.

## Foul Up 3 or Defend

A popular movement in basketball has been to foul the opposing team when the fouling team is leading by 3 points. The logic behind this move is that the trailing team will be forced to shoot free throws, then foul the leading team, then get the ball back with less time on the clock. In order to calculate the probability of the leading team winning given either choice, there were 6 inputs:

- Leading Team’s Free Throw %
- Trailing Team’s Free Throw %
- Trailing Team’s 2-point %
- Trailing Team’s 3-point %
- Shot Clock Effect
- Leading Team’s chance of winning in overtime

I had to make some assumptions to make the calculations doable. First, I decided to ignore the possibilities of offensive rebounding and turning the ball over. Because I am leaving these elements out, the real probability of each decision will be slightly inaccurate, although the rebounds and turnovers should cancel each other out a little. In addition, I decided to assume that a team down 1 point would shoot a 2-point attempt to try to win and that a team down 2 points would shoot a 3-point attempt to win.

### Fouling

In the diagram below, Team A represents the leading team, while Team B represents the team that is down by 3. As seen below, there are 2 cycles of free throw shots followed by one final shot for Team B to win or go to overtime. Because the final shot would come under greater time constraints, I assumed that the field goal percentage for those shots would likely be lower than normal. The “Shot Clock Effect” variable indicates how many percentage points the probability of making a field goal falls when under more difficult time constraints. For example, a shot clock effect of -5% means that a shot has a 5% lower chance of being made if there is less time left.

Note: Colored tiles that denote game result are for Team A (The green tiles mean that Team A would have won)

### Defending

Analyzing the situations that could happen if the leading team opts to play defense instead of fouling is very simple. I assumed that the trailing team would hold the ball for the final shot and shoot a 3 to attempt to force overtime. Therefore, the trailing team’s chance of winning is equal to the chance of making the 3-point attempt times the probability that they win in overtime. In the other scenarios, the leading team (Team A) wins.

### Advantage

Using NBA and College Basketball averages for the statistical categories included in the probability calculations, we can determine if there is a league-wide advantage for either fouling up 3 or defending. The NBA average free throw percentage was 77.5% in 2021-22, and the chance of winning in overtime was set to 50%. The NBA average 2-point percentage during shots that were taken with under 1 minute left in the game when leading or trailing by 5 points was about 43%, and the average 3-point percentage was about 25%. Lastly, I decided to set the Shot Clock Effect to -5%, meaning that a 2-point shot under greater time constraints would have a probability of 38% of being made.

As seen by the graph above, there is no obvious advantage to either strategy. Because the margin is very small, it is likely that accounting for offensive rebounds and turnovers could tilt the advantage to being more towards one side. Nonetheless, the small margin means that the best choice likely depends on the opponent

Because fouling when up 3 depends on both teams making free throws, it is more beneficial for a team to foul if they are playing an opponent with poor free throw shooting, like the Lakers. The Lakers made just 61% of their free throws late in close games last season, so putting them on the line is a great option. Meanwhile, the Hawks were snipers at the line late in games, so defending would be a better option against them. Opponent efficiency also matters, because it is better to foul and hope for a miss at the line instead of allowing a high probability shot.

In game 3 of the Celtics-Bucks series in the past postseason, the Bucks found themselves up 3 points with fewer than 10 seconds remaining in the game. Bucks guard Jrue Holiday opted to foul Marcus Smart, putting the Celtics on the free throw line with 2 shots with little time left. In this situation, Smart decided to intentionally miss the free throw, allowing the Celtics to get an offensive rebound. Ultimately, the Celtics missed several tip-in attempts and the Bucks walked away with the win. Using the win probability calculations, the Bucks actually would have been better off allowing the Celtics to shoot for the win since the Celtics excel at shooting free throws, but it worked out for them in the end.

I decided to omit the possibility of an intentionally missed free throw from the calculations above in order to simplify them. However, in general it is very difficult to get an offensive rebound off of a free throw, so it would not make a huge difference.

## Quick 2 or a 3 for the Tie

There are many occasions where a team finds themselves down by 3 points with the ball. In these tense moments, there are people that prefer to take a 3-point attempt and go for the tie, while other believe that the best option is to go for a quick 2-point attempt and foul. The debate over the best decision is fierce, but the math points in one clear direction.

### Quick 2

Advocates for shooting a quick 2 when down 3 points argue that a team should opt to attempt a higher percentage shot, then foul and get another chance to tie or win the game. Therefore, the scenarios begin with the trailing team (Team A) making or missing a 2-point shot. Afterwards, they must foul Team B (the leading team). The outcomes of the shot attempt and the free throws impact whether Team A lost or can go for a win or tie. Again, the second shot attempt for Team A is more difficult due to less time remaining on the clock, so the chances of making the shot are lower. The exact scenarios are drawn out below.

### 3 for the Tie

While the outcomes after a quick 2-point attempt are complicated, the possibilities after a 3-point attempt for the tie are easy. I assumed that teams choosing to go for a 3 would be taking the final shot of regulation, meaning Team A’s chance of winning is equal to the chance of making the 3-point attempt multiplied with the probability of winning in overtime.

### Advantages

Using the same process as previous sections, here are the NBA and college league-wide win probabilities for each decision.

In both the NBA and college basketball, there is an easy choice to be made: go for the tie. Many coaches overcomplicate things by trying to make 2 shots with the possibility of overtime when they can instead just take one shot and force overtime. Of course, context is important. If the defense is fiercely guarding the perimeter and allowing an easy layup opportunity, it is probably better to take the easy layup.

In the 2022 NBA Playoffs, there were two games in which a team had the option to take a quick 2 or go for the 3 to tie. In Game 5 of the Grizzlies-Timberwolves first round series, the Timberwolves found themselves down 3 with little time remaining. They chose to shoot a 3, and Anthony Edwards made the shot. Unfortunately, the few seconds the Wolves left on the clock were too many as Ja Morant scored a buzzer-beating layup to win the game. Nonethless, the Timberwolves made the right choice by going for 3.

A different choice occurred in Game 4 of the Grizzlies-Warriors series. The Grizzlies were down by 3 with under a minute left, but they chose to shoot a layup instead of attempting a 3. Although Dillon Brooks made the layup, the Warriors made their free throws and the Grizzlies were forced to shoot a low percentage shot from the perimeter that didn’t go in, giving the Warriors a big win.

## 3 to Win or 2 to Tie

The final late game situation I will cover is when a team has the ball down by 2 points. In this situation, the trailing team can either take a 3 point attempt to win the game, or they can go for a 2-point shot to force overtime. The math for this choice is very simple and gives a clear answer.

### 3 to Win

If a team is down 2 and shoots a 3-point attempt for the win, the result of the game rests on the outcome of that shot. If the shot is made, the team wins. Otherwise, they lose. Therefore, the win probability is equal to the chance of making the shot (assuming this is the final shot of the game).

### 2 to Tie

If the trailing team instead opts to go for a 2, the team needs to both make the shot and win in overtime. Therefore, the win probability for this choice is equal to the chance of making the 2-point attempt multiplied by the chance of winning in overtime.

### Advantage

The math for the optimal choice is very simple for this case. If the team shoots a 3, the chance of winning is equal to the 3-point percentage. If they shoot a 2, the chance of winning is the 2-point percentage times the overtime win probability. As seen below, it is better to go for the 3 to win in both the NBA and college, in general.

However, this suggestion changes based on the relative 2-point and 3-point efficiencies for a team. For example, the Clippers had a 45% 3-point percentage in close late game situations last season, so they would be foolish to not take this 45% chance to win. On the other hand, the Cavs shot just 15% in similar situations, so they would be better off shooting a 2 and then trying to win in overtime than forcing a very low percentage shot.

As a side note, it is important to know that the sample of shots is pretty small because I tried to isolate shot attempts that occurred late in close games, so that is why some of the percentages are very extreme.

One real-life example of this choice happened in Game 6 of the first round Mavericks-Jazz series. The Jazz had the ball down by 2 with just a few seconds remaining. Ultimately, Bojan Bogdanovic shot a 3 for the win that just missed, clinching the series win for the Mavericks. Using the Jazz’s regular season late game shooting stats, this was an uncommon situation in which they actually may have been better off going for a 2 to tie and then trying to win in overtime because their 3-point percentage was so low. However, in general it is better to go for the 3 to tie.

## Conclusion

Late game decision-making is an important part of all basketball coach’s and player’s jobs. They need to know how their teams’ win probabilities change with each decision they make. When a team is down by 3 with the option to shoot a 3 for the win or to go for a quick 2, it is better to shoot the 3 to tie, in general. If a team is down by 2, they should usually try to design a play that gets an open 3 for the win instead of going to the rim and trying to tie. Finally, a coach must know how well both his team and the opponent shoot free throws in order to determine if fouling up 3 is a good decision or not. Late game decisions could determine if coaches or players keep their jobs, so knowing the probabilities is crucial.