The gambler's fallacy is the mistaken belief that a random outcome becomes more or less likely because of what happened before. It often appears in coin flips, dice rolls, roulette, sports streaks, and lottery choices. The idea matters because it can lead people to make risky decisions based on patterns that are not real.
Good statistical thinking separates short-term streaks from true changes in probability.
For independent events, the probability of the next outcome does not change after a streak. If a fair coin lands heads 6 times in a row, the probability of heads on the next flip is still 1/2. The law of large numbers says long-run averages tend to get closer to expected probabilities, but it does not force future outcomes to cancel past streaks.
The hot-hand fallacy is a related error where people assume a streak means success is now more likely, even when there is no evidence the probability has changed.
Key Facts
- Independent events do not affect each other: P(next outcome | past outcomes) = P(next outcome).
- For a fair coin, P(heads) = 1/2 and P(tails) = 1/2 on every flip.
- For a fair six-sided die, P(rolling a 6) = 1/6 on every roll.
- The probability of 6 heads in a row is (1/2)^6 = 1/64.
- After 6 heads in a row, the probability of another head is still 1/2, not less than 1/2.
- The law of large numbers describes long-run relative frequency, not a short-run correction: observed proportion approaches expected probability as n becomes large.
Vocabulary
- Gambler's fallacy
- The mistaken belief that a random event is due to reverse after a streak, even when each trial is independent.
- Independent events
- Events are independent when the outcome of one event does not change the probability of another event.
- Conditional probability
- Conditional probability is the probability of an event given that another event has already occurred.
- Law of large numbers
- The law of large numbers states that the average result of many independent trials tends to approach the expected value.
- Hot-hand fallacy
- The hot-hand fallacy is the mistaken belief that a person or process is more likely to keep succeeding just because of a recent streak.
Common Mistakes to Avoid
- Thinking tails is due after several heads is wrong because a fair coin has no memory and each flip still has P(tails) = 1/2.
- Confusing a rare streak with an impossible streak is wrong because unlikely sequences can happen, especially when many trials are observed.
- Using the law of large numbers to predict the next trial is wrong because it describes long-run averages, not a forced correction on the next outcome.
- Assuming every streak means a changed probability is wrong because a streak can occur by chance unless there is evidence that the underlying process changed.
Practice Questions
- 1 A fair coin lands heads 5 times in a row. What is the probability that the next flip is tails?
- 2 A roulette wheel has 18 red spaces, 18 black spaces, and 2 green spaces. If black occurs 4 times in a row, what is the probability that the next spin is red?
- 3 A basketball player makes 6 shots in a row. Explain what information you would need before deciding whether this is evidence of a hot hand rather than random variation.