Anyone interested in sports probably knows the concept of ‘being in form’. When a player or a team performs exceptionally well over a certain period, we intuitively expect that streak to continue. Whether it is a football team winning five matches in a row or a striker scoring in consecutive games, supporters and pundits alike anticipate the next success. But is this assumption mathematically justified? Or are we merely gaslighting ourselves?

In 1985, Thomas Gilovich, Robert Vallone, and Amos Tversky (GVT) published a groundbreaking paper that shattered this sporting illusion: The Hot Hand in Basketball: On the Misperception of Random Sequences. Analysing shooting data from the Philadelphia 76ers, they investigated whether a player who had just hit a few shots in a row had a higher probability of scoring the next one (the definition of a ‘hot hand’).

GVT’s conclusion was brutal for sports romanticists: the hot hand does not exist. They argued that basketball shots should be viewed as a sequence of independent, random events where each shot has a fixed probability of success, determined by a player's general skill and court position. A ‘streak’ is simply a statistical inevitability within a large data set. If you roll a fair die thousands of times, you are likely to eventually roll four consecutive sixes. This, however, does not increase your probability of throwing a six on the fifth roll.

Humans have a natural, evolutionary tendency to recognize patterns, even where they do not exist. While this cognitive skill is crucial for understanding the world, learning from experience, and predicting threats, GVT argued it also tricks us into seeing 'streaks' and 'form' where there is only pure, unadulterated randomness.

Case closed? Not quite!

In 2018, economists Joshua B. Miller and Adam Sanjurjo turned the academic world upside down by proving that GVT had fallen into their own cognitive trap. They uncovered a subtle but devastating selection bias inherent to finite sequences. To understand this, consider a fair coin tossed exactly four times. There are sixteen possible outcomes (2^4). If we look at every sequence, select only the coin tosses that directly follow a 'Heads', and calculate the success rate of throwing another 'Heads' within that specific sequence, a mathematical miracle occurs.

Intuitively, you would expect the average probability to be 50%. However, if you calculate the percentage per sequence and average them over the fourteen sequences that contain data, the expected value is actually only 40.5%. Because a player's career is strictly finite, their shooting data suffers from this exact streak selection bias.

This realization completely inverts GVT's original conclusion. If a player’s expected probability after a hit should mathematically drop to 40.5% due to the finiteness of the sequence, but GVT’s data showed they continued to shoot their baseline average of 50%, the player must have significantly outperformed the statistical baseline. While basketball players obviously do not have a uniform 50/50 chance of making a shot, Miller and Sanjurjo proved that this selection bias shifts the baseline for any fixed shooting percentage. The hot hand is real; it was just masked by a flawed null hypothesis. 

Yet, in actual basketball games, an even more complex variable is at play: endogeneity. When a striker or basketball player gets 'hot', the opposing team notices. They adjust their strategy, tightening the defense or applying double-teaming. The fact that a player maintains their average shooting percentage while facing a significantly higher defensive pressure is, in itself, economic proof of an increased capability. To eliminate this confounding variable, Miller and Sanjurjo took their model to the NBA Three-Point Contest, an environment with finite streaks but entirely without defense. The results were undeniable: after controlling for selection bias, players on a streak showed a massive, statistically significant increase in their shooting percentage.

So, next time you watch a game and feel the absolute certainty that a player is going to score their next shot, do not let the traditional psychologists tell you that your brain is playing tricks on you. Your intuition was never flawed. It turns out that human intuition was actually doing some incredibly sophisticated, subconscious econometrics all along, outsmarting even the most prominent cognitive researchers for over thirty years.