The Eastern Conference Finals between the Celtics and Heat has been a tightly contested series, with each game being unpredictable. In Game 7, the game that would decide which team faces the Warriors in the Finals, the Heat found themselves down just 2 points with less than 30 seconds left in the game. Marcus Smart missed a shot at the rim, leading to a Jimmy Butler rebound. Butler took the ball down the court and with 18 seconds left in the game, he pulled up from 3 in transition for the win. After the heroic shot attempt hit the front of the rim, Al Horford grabbed the defensive rebound and passed to Marcus Smart, who sealed the game by hitting both free throws following an intentional foul. We know now that Butler’s final shot didn’t fall, but using the win probability if he chose to do something else can tell us whether Butler’s shot attempt was the right decision.

## Butler’s Choices

To simplify this, I will say that Jimmy Butler had 4 choices of what he could have done as he crossed half court during transition. His first option was to take the early pull-up 3, which he did. Another option would have been to drive at Al Horford, who was backpedaling while only Jaylen Brown was the only other defender who would have helped him. His third option would have been to drive, force help from Jaylen Brown, then pass out to Victor Oladipo on the left wing for an open catch and shoot three point attempt. Lastly, Butler could have held the ball and the Heat could’ve taken the final shot, eliminating the possibility of Boston getting another offensive possession. Using season and series stats, we can find the win probability associated with each of these decisions.

## Scenario 1: Early Pull-Up 3 (What Actually Happened)

In the first scenario, Jimmy Butler takes the early pull-up transition 3 with 18 seconds left in the game. There is a chance that he makes or misses the shot. If he makes the shot, the Heat go up 1 and the Celtics have an opportunity to score and win. If Butler misses, though, the Heat have to intentional foul. With the Celtics up 2 points with a Butler miss, the Heat need the Celtics to miss at least 1 free throw. If the Celtics make 1 free throw, then the Heat are down 3 and need to make a 3-pointer to tie and after that win in overtime. If the Celtics make 2 free throws, the Heat have the option to go for the tie with a 2-point attempts or go for the win with a 3-point attempt.

The outline makes some assumptions, like there won’t be an offensive rebound off the free throw, in order to make the calculations easier. In order to calculate the chance of winning, we first need the probabilities of:

- Butler making pull-up 3
- Celtics scoring on 1 possession
- Boston making 0, 1, or 2 free throws (out of 2 attempts)
- Heat scoring a 2 with time expiring
- Heat scoring a 3 with time expiring
- Heat winning in overtime

Jimmy Butler’s pull-up 3-point percentage during the 2021-22 regular season and 2022 playoffs was 25.8% as hit hit 17 of 66 shot attempts. The chance of the Celtics scoring on 1 possession can be calculated using game-level stats. The Celtics had about an estimated 98.6 possessions (85 FGA + 0.44*24 FTA + 13 TOV – 10 OREB). They scored at least 2 points on about 41.6 of these possessions (FGM + (FT%)^{2} * 0.44 * FTA), so they scored on about 42.2% of possessions.

The Celtics were shooting 81% on free throws during the playoffs, so their chance of making 2 in a row is equal to 0.81^{2} which is 65.6%. The chance of making 1 free throw is 30.8% and the chance of missing both is 3.6%.

The probability of the Heat making a 2-point shot or a 3-point shot with time expiring can be estimated using field goal attempts that came when the shot clock was between 0 and 4 seconds. Through the regular season and playoffs, the Heat made 40% of 2-pointers and 28.4% of 3-pointers in this scenario. However, Miami could also turn the ball over if they get the last shot. The Heat’s turnover rate in the playoffs is 12%, so this is the chance they turn the ball over. Therefore, Miami gets a shot on 88% of possessions, so they score a two on 40% times 88% = 35.2% of possessions when they want a 2 and they score a three on 28.4% times 88% = 25.0% of possessions when they want a 3. Lastly, we can assume that both teams have an equal chance to win in overtime, so the win probability is 50%.

Now, we can calculate Miami’s win probability for this shot. The simplest way Miami wins is if Butler hits the pull-up 3 and the Celtics fail to score. The chance of this happening is 14.9%. If Butler misses, Miami can win if Boston misses 1 free throw, they score a 3-pointer, and they win in overtime. They can also win if Boston misses both, they score a 2-pointer and win in overtime or they score a 3-pointer and win outright. The chance that Butler misses the pull-up (74.2%), Boston misses 1 free throw (30.8%), Miami scores a late 3-pointer (25.0%), and the Heat win in overtime (50%) is equal to the product of the four probabilities, which is 2.9%. Lastly, if Butler misses (74.2%) and the Celtics miss both (3.6%), Miami has the option of going for 2 to tie (35.2%) or a 3 to win (25.0%). We don’t know whether they would have gone for 3 or 2, but given that most of the guys on the floor had been playing nearly the entire game, we will assume Miami would take the 3 to win. The chance of Miami winning this way is 74.2% x 3.6% x 25% = 0.7%. Adding up the 3 winning possibilities (14.9% + 2.9% + 0.7%) shows that Miami had a 18.5% chance of winning with Butler’s pull-up 3.

Chance of Winning: 18.5%

## Scenario 2: Butler Drives and shoots a 2

Something that nearly everyone watching the game saw is that Butler had a great opportunity to drive instead of taking the pull-up 3. Many of the probabilities from the previous section carry over, but the chance of Butler making a driving shot can be estimated by his true shooting percentage on drives during the regular season and playoffs, which was 61.6% (using TS% helps to account for possibility Butler gets fouled on the drive). The scenarios for Butler missing the shot remain the same, but they are different for when Butler makes the shot. If Jimmy Butler makes a driving shot, the Heat need to force a stop (57.8%) then win in overtime (50%). The chance of winning this way is 61.6% x 57.8% x 50% = 17.8%.

The chance of Butler missing (38.4%), Boston making 1 of 2 free throws (30.8%), Miami hitting a 3 (25.0%), and then winning in overtime (50%) is equal to 1.5%. The chance of Butler missing (38.4%), Boston missing both free throws (3.6%), and Miami hitting a 3 to win (25.0%) is 0.3%. Therefore, the chance of Miami winning with a Butler drive is equal to 17.8% + 1.5% + 0.3% = 19.6%.

Chance of Winning: 19.6%

## Scenario 3: Butler passes to Oladipo for open 3

This scenario is very similar to scenario 1 in that a make gets Miami 3 points and a miss goes through the same set of steps. However, the chance of Oladipo making an open 3 is different than the probability that Butler makes a pull-up 3. Oladipo went 12/41 on open 3’s during the regular season and playoffs, which is 29.3% (surprisingly only slightly higher than a Butler pull-up 3). There is also a chance that Butler turns the ball over. During the playoffs, Butler turned the ball over on just 10/244 (4.1%) of his drives. Therefore, the chance that Butler successfully passes to Oladipo and Oladipo makes the shot is 95.9% x 29.3% = 28.1%. The chance Miami wins with an Oladipo made 3 then a stop equals 28.1% x 57.8% = 16.2%.

The chance of Oladipo missing or Butler turning it over, Boston making 1/2 free throws, and then Miami winning (same calculation as the previous scenarios) is 2.8% and the probability of Miami not scoring, the Celtics missing 2, and Miami winning is 0.6%. The chance of Miami winning with Butler driving and passing to Oladipo is approximately 19.6%.

## Scenario 4: Miami holds for final shot

This scenario holds the easiest calculations. Since Miami would hold for the final shot, Boston does not get the opportunity of another possession. Therefore, we only have to account for Miami’s decision.

If the Heat held for the final shot, they could take a 2-pointer to tie then play overtime or attempt a 3 to win. If Miami held for the last shot, took a 2-pointer, and played overtime (we’ll call this scenario 4a), their win probability would equal 35.2% x 50% = 17.6%. However, if Miami took a 3 to win at the buzzer, they would have a 25.0% chance to win (scenario 4b) since all they have to do is not turn it over then hit a 3 with 0-4 seconds left in the shot clock.

## Win Probabilities

Here is the chance of winning for each scenario:

- Scenario 1 (Butler early pull-up 3): 18.5%
- Scenario 2 (Butler drives): 19.6%
- Scenario 3 (Butler drives and passes to Oladipo): 19.6%
- Scenario 4a (Miami hold for last shot, 2-pt attempt): 17.5%
- Scenario 4b (Miami hold for last shot, 3-pt attempt): 25.0%

From these win probabilities, there is a clear winner. The best option for Butler would have been to hold the ball for a Miami 3 and prevent the Celtics from having a chance to answer. These calculations require lots of assumptions, so obviously they are not perfect. However, the fact that holding for a last second 3 is over 5 percentage points better than the next best option indicates it should have been the obvious choice. When in the midst of an intense game, though, it is impossible to know the chances of winning after each choice, making Butler’s decision understandable. I think that the biggest lesson from this exercise is that teams should almost always try to hold for the final shot. Even though the actual shot may have a lower chance of being made, it outweighs that fact shooting early allows the other team an extra possession. Butler’s decision making was not wrong in that he took a low probability shot, but rather that he left too much time on the clock for the Celtics to answer even if he made that shot.

These calculations also help to show that at the end of a game, often times teams need to shoot a 3. It is common to hear NBA announcers say that a team doesn’t need a 3 in the final seconds of a game, but as shown by the difference in win probability between scenario 4a (shooting a 2 at the end of the game) and 4b (shooting a 3 at the end of the game), the lower value 2-point shot required both a make and a win in overtime. Even when its not the final shot, a 3-pointer is usually the right choice. For example, imagine a team is down 3 points with less than 30 seconds remaining in a game. It is unlikely that a team will make a 2, the opponent will miss a free throw, then the team will make another 2 to force overtime. Rather, it is more likely that the team will just make a 3-point shot and force overtime. Using the probabilities from the Heat and Celtics scenario, the chance of winning while down 3 and going for a 2 is about 8.1% (depends on chance of making last second 2, chance of making last second 3, and opponent’s FT%) while the chance of winning by going for a 3 to tie is approximately 12.5%. In conclusion, Jimmy Butler shouldn’t have shot the ball and allowed Miami to hold for the final shot (which should be a 3), and in late game situations where a team is unsure whether to attempt a 2 or a 3, they should almost always shoot for 3.