# The Hot Hand Fallacy

Taxonomy: Logical Fallacy > Formal Fallacy > Probabilistic Fallacy > The Hot Hand Fallacy1

Sibling Fallacy: The Gambler's Fallacy

 A gambler has had a streak of good luck. Therefore, the gambler is "hot" and the good luck will continue at a probability greater than chance. A gambler has had a streak of bad luck. Therefore, the gambler is "cold" and the bad luck will continue at a probability greater than chance.

### Exposition:

This fallacy is committed every day in casinos around the world, whenever a gambler thinks he's "hot". When gamblers are on winning streaks, and keep betting or increasing their wagers to take advantage of their good luck, they commit this fallacy. Despite its name, gamblers also commit the fallacy when they think that they're "cold", and stop betting or decrease their wagers because they're on a losing streak. This is still the "hot hand" fallacy, because the logical mistake is the same.

The fundamental error is the same as in its sibling fallacy, the Gambler's Fallacy, namely, the failure to appreciate statistical independence. Two events are independent when the occurrence of one does not change the probability of the occurrence of the other2. For instance, if you roll a die and flip a coin at the same time, then whether the coin comes up heads or tails does not affect the number that comes up on the die. So, a roll of a die and flip of a coin are statistically independent events. In a fair gambling device, such as an unloaded die, the number that comes up in one roll of the die does not affect the number that comes up in a different roll, that is, the rolls of the die are probabilistically independent. This is what is meant by a "fair" gambling device.

Both the Hot Hand Fallacy and the Gambler's Fallacy are tempting mistakes to commit. How can you avoid doing so? One way to keep statistical independence in mind is by reminding yourself that fair gambling devices do not have memories. Coins and dice and roulette wheels do not remember what they've done in the past. Also, just as a fair gambling device does not remember its own past, it does not remember a gambler's past. So, a gambler's odds of winning a current bet are not affected by whether he or she has won or lost previous ones.

### Exposure:

• Can you guess which of the following is a random sequence of coin flips: "H" is heads, "T" is tails?
1. H, T, T, H, H, H, T, T, H, H, H, H, H, T, T, H, H, T, T, H
2. H, T, T, H, H, T, T, H, H, T, H, H, T, T, H, T, H, T, T, H

The first sequence is random, despite the fact that there is a run of five heads in a row, and there are a total of a dozen heads but only eight tails. The second sequence is simply the first reversed, and two of the heads changed to tails in order to remove the five head streak and balance the numbers of heads and tails to ten apiece. If you thought the second sequence was random and the first not, you were probably making the mistake of thinking that a short random sequence should resemble a long one, that is, should have an equal number of heads and tails in this case. This has been called "the law of small numbers"3, which is not a "law" at all, but the mistaken belief that small random samples should closely resemble large ones.

Psychological research has shown that people tend to underestimate how "streaky" random sequences can be. As a result of the belief in the "law" of small numbers, people may be more inclined to be surprised by streaks such as the five heads in a row in the first sequence, above. This may lead them to think that some mysterious luck or power is at work when a streak occurs that either benefits or harms a gambler; in the former case, he is "hot" and in the latter "cold".

We all are lucky on some occasions and unlucky on others, but the problem with luck is that it can only be recognized in hindsight. The fact that you've been lucky or unlucky in the immediate past is no reason to think that the good or bad luck will continue in the future.

• Ironically, the Hot Hand Fallacy and its closely-related sibling, the Gambler's Fallacy, can lead to contrary expectations about what will happen next. For instance, suppose that someone bets on a "lucky" number, and wins several times in a row. The Gambler's Fallacy predicts that the lucky number will be less likely than chance to come up on the next bet, since it's necessary for the odds to even themselves out to make up for the excess occurrences of the lucky number. In contrast, the Hot Hand Fallacy predicts that the lucky number is more likely to come up, since the bettor is on a winning streak. This means that both predictions cannot be true, despite the fact that many gamblers probably have committed both fallacies, even on the same day, though not at the same time. So, these two forms of argument cannot both be cogent, and in fact both are uncogent. Both are uncogent for the same reason: fair gambling devices do not remember the past.
• The so-called "hot hand" in basketball and other sports may be a different phenomenon than that discussed here, though it is still controversial4. Since basketball and other sports are games in which skill plays a role, it's possible that psychological factors may lead players to hit or miss in streaks greater than would be accounted for by chance. This is not the case for most gambling games, such as craps or roulette, which are games of pure chance. Some gambling games do have elements of skill involved, including such card games as poker and blackjack. A well-shuffled deck of cards has no memory, but the blackjack player does and, if he can remember the cards that have been dealt, he may be able to gain a slight advantage over the dealer. For this reason, one must be careful about applying gambling fallacies to games that involve skill, since the events of the game may not be probabilistically independent.

Notes:

1. James Sundali & Rachel Croson, "Biases in Casino Betting: The Hot Hand and the Gambler's Fallacy", Judgment and Decision Making, Vol. 1, No. 1, (7/2006).
2. Roger Porkess, The Harper Collins Dictionary of Statistics (1996), see under: "independent events". In probability theory, event E1 and event E2 are independent if and only if P(E1) = P(E1|E2), that is, the probability of E1 occurring is the same as the probability of it happening if E2 occurs. For more on probability theory, see the entry for Probabilistic Fallacy.
3. Amos Tversky & Daniel Kahneman, "Belief in the Law of Small Numbers", in Judgment Under Uncertainty: Heuristics and Biases (1985), Kahneman, Paul Slovic & Tversky, editors, pp. 23-31.
4. Thomas Gilovich, How We Know What Isn't So: The Fallibility of Human Reason in Everyday Life (1991), pp. 11-17.

Revised: 11/18/2020