^{ Gary Smith July 28, 2023 12 Probability }

# Sabrina Ionescu’s Hot Hand

_{When basketball players hit a "streak," does that elevate the probability of success?}

_{ Gary Smith July 28, 2023 12 Probability }

Most people believe that athletes sometimes get “hot” or “cold” with their performance elevated or depressed temporarily. For example, Purvis Short, who scored 59 points in an NBA game, said, “You’re in a world all your own. It’s hard to describe. But the basket seems to be so wide. No matter what you do, you know the ball is going to go in.” Similarly, during a timeout in a 2015 game, LeBron James told his teammates to pass the ball to Kevin Love, explaining after the game that, “He had the hot hand, I wanted to keep going to him.”

On the other hand, statisticians tell us that streaks are likely even in random coin flips that have a rock-steady 50-50 chance of heads or tails. For example, in 10 coin flips there is a 46% chance of a streak of either four or more heads in a row or four or more tails in a row. A streak—in coin flips or sports—doesn’t necessarily mean that the probability of success is elevated or depressed.

Three psychologists—Thomas Gilovich, Robert Vallone, and Amos Tversky—argued that the common “hot hands” belief is a prime example of how people misperceive random sequences by creating causal explanations for noise. They analyzed data from Philadelphia 76ers NBA home games during the 1980-1981 season and concluded that streaks of hits or misses happened no more often than do streaks of heads or tails in coin flips.

## The “Hot-Hands Myth”

The impact of their paper was so great that the widespread belief in the hot hands became known as “the hot-hands myth.” However, several confounding factors make these basketball data not at all like coin flips: during games, basketball shots are taken from very different locations at very different times with varying degrees of defensive pressure.

I looked at two sports, horseshoes and bowling, that are more like coin flips and found evidence of, if not hot hands, at least warm hands. Another appealing set of data is the 3-point shooting contest held during the NBA and WNBA All-Star Weekends.

In the NBA contest, participants have 60 seconds to take five unguarded shots from each of five locations outside the 3-point line. (The past three contests included two long-distance shots that are nearly six feet beyond the 3-point line.) The fact that the success probabilities might vary somewhat from station to station makes outstanding performances even more remarkable.

The greatest performance so far was in 1991 when Craig Hodges made 21 of 25 shots, including an astonishing 19 in a row. To evaluate whether this could be explained as simple luck, we have to take into account the fact that the NBA 3-point contest has been around since 1986 with multiple participants and multiple rounds each year. With so many opportunities for something remarkable to happen, is it really remarkable when something unusual does happen?

I looked at the history of the contest and concluded that it was not particularly remarkable that someone would make 21 of 25 shots. Hodges’ streak of 19 in a row, however, was “too incredible to be explained by luck or cherry-picking. He was hot.”

Another extraordinary performance happened in 2015 when the Golden State Warriors reported that Steph Curry made 94 out of 100 3-point shots in practice, including an astonishing 77 in a row. In 2020, Curry did even better, making 105 in a row. It is difficult to assess the statistical significance of these feats because we don’t know how many people have been practicing this many 3-point shots since the NBA introduced them in 1979, nor do we know what their individual success probabilities are.

Jay Cordes and I made some calculations based on a coin-flip model with conservative assumptions. The coin-flip model assumes that a specified number of people essentially flip a coin with a specified probability of heads a specified number of times. If the probability is very low that a remarkable feat (like 105 heads in a row) would happen, then this is evidence against the coin-flip model—and evidence that something special happened.

We calculated that, if, every day of every year for 50 years, 500 players with 70% accuracy take 500 consecutive 3-point shots, the probability that someone will make 105 in a row is a very, very low 0.000000059. Even if 100 players with 80% accuracy take 500 consecutive 3-point shots every day for 50 years, the probability of a 105-shot streak is still only 0.0049. We concluded that “it is very hard to explain Curry’s 105-shot streak as just another lucky day on the basketball court. He got gloriously red hot.”

## More Than Sheer Luck?

Now we have WNBA star Sabrina Ionescu’s widely reported performance in this year’s WNBA 3-point contest: 25 of 27, including 20 in a row. Comparisons of male and female performance are problematic because the WNBA ball is smaller than the NBA ball and the 3-point line is closer to the basket but the question here is whether Ionescu’s performance is evidence that she got hot or was just lucky. The analysis is challenging because there are only three years of data for the 27-shot version of the contest.

Making a statistical analysis even more challenging, the historical record is muddied by the fact that news reports generally do not say which shots were made or missed, only the player’s total score based on a complicated scoring system: the two distance shots are worth 3 points each; four of the five racks have four 1-point balls and one 2-point “money ball;” and the fifth rack has five money balls.

When I assessed Hodge’s performance, I had shot-by-shot data for nineteen 25-shot rounds he had participated in, so I was able to calculate the probability that a random ordering of the 475 shots he took during these 19 rounds would yield one round that was so incredible.

I can’t take that approach here since this is the only 3-point contest that Ionescu has competed in, so I used the coin-flip model. The top WNBA (and NBA) players make roughly 45 to 50 percent of their 3-point shots in game situations. Ionescu’s current 3-point percentage in regular-season games is 44.3. The players’ success probabilities in the All-Star contest with no defenders in sight are surely higher than during regular season games but not nearly as high as the 80%-90% free-throw accuracy for elite shooters. Historically, NBA players averaged 55% accuracy in the 25-shot contests, so I will use a range of 50% to 60%.

Table 1 compares the coin-flip probabilities for Hodges and Ionescu, considering just the single occasion when they did so well. As I noted earlier, Hodges’ 19-shot streak was more impressive than his making 21 of 25 shots. Ionescu’s performance was even more impressive, particularly if we acknowledge that it included two long-distance shots with surely lower success probabilities.

**Table 1 One Person’s Probability of Doing this Well (or Better)**

Hodges’ 25 shots Ionescu’s 27 shots

Coin-Flip Accuracy Make 21 Streak of 19 Make 25 Streak of 20

0.50 0.000455 0.000007 0.000003 0.000004

0.55 0.002309 0.000044 0.000025 0.000027

0.60 0.009471 0.000209 0.000179 0.000141

We have to put the Table 1 probabilities into context by considering how many players have had opportunities to do this well. In the NBA, a total of 124 players participated in the 25-shot contest (taking into account the qualifying and championship rounds). In the WNBA, 24 players have participated in the 27-shot contest. Table 2 shows the probabilities that at least one player would do as well as Hodges and Ionescu did.

**Probability that At Least One Player Would Do This Well**

Hodges’ 25 shots, 124 players Ionescu’s 27 shots, 24 players

Coin-Flip Accuracy Make 21 Streak of 19 Make 25 Streak of 20

0.50 0.054870 0.000867 0.000072 0.000096

0.55 0.249223 0.005441 0.000600 0.000648

0.60 0.692721 0.025585 0.004287 0.003379

Table 2 shows that even if every player who has participated in the WNBA’s 3-point contest had a constant 60% chance of making a shot, the probability that someone would do as well as Ionescu did in the 2023 contest is very small. The more likely explanation is that she got hot.

On the other hand, Curry’s 77-shot and 105-shot streaks are orders of magnitude more impressive than Hodges’ and Ionescu’s streaks, even taking into account all of the people who may have practiced 3-point shots over the past 50 years.

All of these performances support the common perception that athletes do sometimes get hot—not that their chance of success is 100% but that it is temporarily elevated above their normal probability.