Let’s jump in our DeLorean go back in time – not to 1984, but 1985. And instead of picking up a copy of Grays Sports Almanac, we’ll grab a paper published by Gilovich, Vallone, and Tversky (GVT) that year.

It’s called The Hand in Basketball – On the Misperception of Random Sequences, and it’s all about whether basketball players really have “the hot hand”.

That is, do players hit a “hot streak” – a period of time where they’re playing better than usual, and they are more likely to successfully make baskets than they otherwise would be.

Or, are such streaks just down to plain old-fashioned randomness?

(I guess the title of the paper is a bit of a spoiler…)

## You cannot be serious

Most people believe in the hot hand. And with good reason – think of all the evidence out there on how psychological states like flow and confidence can influence performance. This would all point in the direction of hot streaks being possible. Plus anyone who has ever played a sport or game has experienced a hot hand, a period of time where you’re just “on it”. And even if not, we’ve all seen examples of it on TV.

Besides, we know that the “cold hand” is real. Players routinely have games, or parts of games, where they can’t seem to do anything right. It happens in every sport. Anything from missing an open goal, a hostile crowd, or a bad call from the umpire can put players on tilt, knock their confidence, and cause them to under-perform. So if there’s a cold hand, there’s gotta be a hot one too. It’s how the universe works, yin and yang and all that, right?

Well here’s the counter-point. Take a completely random game like a coin toss. In a normal coin toss, with no tricks, you have no influence on the outcome. Yet, flip enough coins, and you’ll get some streaks. You might get 5, 6, or 7 heads in a row. And you might even feel like you’re doing it – maybe you’ll feel more confident, more focused, and “in the zone”.

But ultimately, you’re being fooled by randomness. This is what Gilovich, Vallone, and Tversky argue happens during a basketball game. When players take enough shots during a game, they’ll get some streaks. And it will feel to them, and look to others, like a hot hand. But in reality, it’s just the ebb and flow of randomness.

So, which idea is right?

## Comparison to randomness

This is a surprisingly tricky thing to study. You’d need to make a comparison to a random sequence of numbers, and see if the players’ performances ever deviated from that.

With a coin toss, this would seem relatively easy. You know the baseline probability is 50% per outcome, assuming there are no tricks at play. With basketball shots, the probability of a successful basket isn’t known. Plus it varies with every shot, depending on things like:

- The skill level of the player
- The skill level of the opponents
- Where the shot was taken from
- The type of shot
- The fatigue level of the player
- The mental focus of the player at that time
- If the player was carrying an injury
- How far into the game it was
- Whether the player was under pressure at the time

…and so on.

So how would you actually work this out?

How do you get to that baseline, the equivalent of the 50% probability for the coin toss? As we’ll soon see, this is a crucial question.

## Recent hits and misses

One one hand, if hot streaks are real, then all this variation and noise shouldn’t matter *too* much. I mean, if they are real enough for people to recognise them, then they should appear in the data despite all that. So the first thing GVT did was just see if players were more likely to make a shot successfully after a previously successful shot (or series of them), or after a previous miss (or series of them).

The baseline was established simply by working out the player’s hit to miss ratio overall. So they didn’t separate based on free throws, shots under pressure, timing, or anything like that. They did the old statistician’s trick of throwing it all in one bucket and just taking the average.

Now here’s another important point to note – these tests were based on data from 9 players in the 48 home games of the Philadelphia 76ers in the 1980/81 season. So we’ve got only a small sample of players here (sports wasn’t as data-focused back in those days).

Here are the overall probabilities they found:

Hit after 2 misses: 56%

Hit after 2 misses: 53%

Hit after 1 miss: 54%

Overall probability of a hit: 52%

Hit after 1 hit: 51%

Hit after 2 hits: 50%

Hit after 3 hits: 46%

As you see, these figures are the wrong way round to what you would expect if hot (and cold streaks) were real.

## A fallacy is born

And thus, the “Hot Hand Fallacy” – the false belief that a successful outcome increases your chance of further successful outcomes – was born. The argument went as follows: The human brain isn’t a statistical computer, and it doesn’t understand randomness very well. So when it sees a string of random outcomes, it assigns agency to them, even though it’s really just noise.

GVT’s idea did have its critics.

“Is nothing sacred?”, Larkey, Smith and Kadane said in their 1989 critique of the fallacy.

“Nope,” GVT effectively replied, in their convincing rebuttal.

The results of GVT’s original study were replicated a few times. Plus the “T” in GVT is Amos Tversky, who would have won a Nobel Prize for his work on cognitive biases, had he not passed away in 1996 (his research partner, Daniel Kahneman, did receive the award). So, despite many doubters outside academia (people in the world of sports didn’t tend to believe it), and some hold-outs within the ivory towers, the hot hand was generally believed to be a fallacy, a cognitive illusion.

And so things remained, for around 30 years or so. Until Joshua Miller and Adam Sanjurjo (let’s call them MS) came along.

## All about that base

MS argue that the hot hand is real, streaks exist, and GVT simply made a mistake in their analysis. And the mistake, they argue, is an ironic one – it has to do with a misunderstanding of randomness.

Let’s go back to that base rate probability I mentioned earlier. In the world of random coin flips, the probability of a head is always 50%. A previous flip of heads will have no bearing on any subsequent flip – so it’s always 50%.

However, when you get outside the world of “standing there, coin in hand, ready to flip”, and into the world of “selecting a sample of flips from a previously completed sequence of coin flips,” your chance of selecting a head will only be 50% if:

- The original coin flips were actually random (which again, we’re assuming here)
- There is no bias in the way you select the flips

MS argue, that selecting flips based on whether they follow a particular outcome (or series of outcomes) *does* introduce bias. In fact, if you look only at the outcomes that follow a flip of heads in a finite sequence of coin flips, the probability of getting another heads is actually lower than 50%.

The more flips there are in the sequence, the closer to 50% it will be, but you still won’t actually get there.

## WTF?

Yeah, that was pretty much my reaction too. But here’s how MS illustrate it:

Let’s say you flip the coin three times. Is your chance of getting a heads higher in a flip, if you already got a heads in the previous flip? There are only a finite number of possible outcomes here, so we can work this out:

HHH

HHT

HTT

HTH

TTT

TTH

THH

THT

First of all, the sequences TTT and TTH are of no use to us. In TTT there is no heads, and in TTH we got a heads, but it’s at the end of the sequence, so we can’t follow it with another coin toss (again – the fact that we’re looking back on a finite sequence of flips is introducing the bias – nothing MS are saying here has any bearing on how future coin flips with turn out).

So, let’s take those two out. That leaves us with 6 sequences we can work with:

HHH

HHT

HTT

HTH

THH

THT

The fact that we only have three flips in each sequence means that the last coin flip only useful as an outcome relative to the second flip. So we have 12 flips of interest here – the first two in each sequence.

So, within each sequence, what are the odds that one of the initial two flips is a heads, which is then followed by another heads?

Let’s see…

Sequence | Heads following heads | Probability |

HHH | 2/2 | First two flips are heads, both followed by heads, so 2 out of 2, probability 1. |

HHT | 1/2 | First two flips are heads, one is followed by heads, so 1 out of 2, probability ½. |

HTT | 0 | First flip is heads, not followed by heads. Probability 0. |

HTH | 0 | First flip is heads, not followed by heads. Probability 0. |

THH | 1/1 | Second flip is heads, is followed by heads. Probability 1. |

THT | 0 | Second flip is heads, but not followed by heads. Probability 0. |

Since we’re assuming pure randomness, each of these sequences is equally likely to happen. That means we can simply take the average of each of these probabilities to get the overall probability that a heads will follow a heads:

(1 + ½ + 1) / 6 = 2.5/6 or 5/12 if you don’t like decimals in your fractions.

That’s less than 50%!

## Small and large numbers

This doesn’t mean that randomness suddenly stops working. It just means that when you have a finite sequence of flips, and you are not taking a random sample of them, but instead only the ones that follow a streak, you’re introducing sample bias into your analysis.

If the sequence of numbers stretched out to infinity, the law of large numbers would take over and the probability would be 50%. But since you’re dealing with a sample, you don’t get that protection.

When MS reanalysed the GVT data with this bias in mind, they did indeed find evidence for the hot hand. In fact, MS not only think the hot hand is real, they think it’s a bigger effect because of another form of measurement error that is common in studies of the hot hand (read more of MS’s thoughts here).

And MS aren’t alone in this – a number of other researchers have chimed in with work supporting the idea that the hot hand is real, and not just a fallacy:

- Revisiting the Hot Hand Theory with Free Throw Data in a Multivariate Framework – Arkes, 2010 – evidence for the hot hand in free-throw situations.
- Hot Shots: An Analysis of the ‘Hot Hand’ in NBA Field Goal and Free Throw Shooting – Lantis and Nesson, 2019. They say they found evidence for the hot hand in free-throws, but not open play.
- Are Points in Tennis Independent and Identically Distributed? Evidence From a Dynamic Binary Panel Data Model – Klaasen & Magnus, 2011 – hot hand effect in tennis.
- Heat Check: New Evidence on the Hot Hand in Basketball – Bocskocky, Ezekowitz and Stein (2014) – 1.2% increase in the odds of making a shot for every shot in a streak.

## Statistical hot hand versus psychological hot hand

It’s also worth mentioning that in all these studies the researchers define a hot hand as a series of successful shot attempts. This makes sense statistically, but it probably doesn’t represent the psychological reality of having a hot hand.

Would your hot hand immediately go cold if you missed a single shot? Probably not. Also, if you have the hot hand, would your newfound confidence lead you to take more difficult shots, in situations when you’d probably play safer if you didn’t have the hot hand? It might be that the sequence of scores is interrupted not because the player lost their hot hand, but because they tried something a little fancy out.

With all the data in sports these days, that’s something you can actually test for, and the study from 2014 I mentioned earlier did indeed find that players who were performing above their average tended to attempt more difficult shots in open play. And this study concludes that there’s evidence for the hot hand – but only after controlling for shot difficulty.

Time could be important too. These studies don’t take the time that each shot was taken into account – and when we’re talking about a psychological state, this seems pretty crucial! Take the following sequence of shots:

101010101

Statistically, there is no streak here, no hot hand. But imagine this sequence of shots happened in the space of a short space of time, and the player wouldn’t ordinarily hit 5 baskets in that time span. Could that not be called a hot hand? If you only define the hot hand in terms of hits in a row, it clearly isn’t. But the player might be running rings around the other team, and getting into shooting positions more easily. If you define the hot hand in terms of hits within a given time period, they now start to look hot.

One thing that doesn’t seem to be in dispute, is that players *believe* in the hot hand, and you might assume they’re more likely to give the ball to hot players. So you might also assume that defenders will try to crowd out hot players. In other words, a player could be hot in the sense that, whatever psychological and physiological changes relating to being hot are active, but this is not reflected in the hit/miss stats because the opposing defence is wise to them. Interestingly, that same 2014 study which found hot players taking harder shots also found that hot players face tougher defence. This is another reason that “hotness” wouldn’t show up if you only look at hit streaks, yet they could still be having an important influence on the game by drawing defending players towards them, and opening up space elsewhere.

Yet another aspect of timing that might be relevant is the breaks in the game. For example, does a hot hand carry over between halves, quarters, or after time outs (or is it more or less likely to)? Should you start a streak from scratch at these points, or allow them to continue?

So, we’re 30 years into this and we still have a lot of open questions. What do you think, is the hot hand real?