Published · · Last reviewed · · 6 min read
English · 한국어
The gambler's fallacy: why a number is never 'due'
A coin has landed heads five times in a row. What comes next? Many people feel that tails is now due. Others feel the opposite: the streak is heads, so keep going. The two intuitions point in opposite directions but share a root: the belief that earlier results influence the next one. For a fair coin the sixth flip is heads with probability exactly 50%. This page confirms that with experiment and calculation rather than intuition, and explains why the belief is so persistent.
First, an experiment: what follows a streak of heads?
The tool below flips a fair coin a million times. It sorts the flips by "after at least 1, 3, 5 or 7 heads in a row" and counts how often the next flip was heads.
The numbers change each run, but every row sits near 50%. After 7 heads in a row the situation arises only a few thousand times, so you may see 49% or 51%; that is small-sample wobble, not a direction. The simulation shows no direction for a simple reason: the code never uses earlier results. A real coin, a draw machine or a random number generator does not use them either.
What "independent" means
Two events are independent if knowing that one occurred leaves the probability of the other unchanged. The sixth flip is heads with probability 1/2 whether or not you know the first five. After five heads, the sequences HHHHHH and HHHHHT are two of 64 equally likely outcomes. HHHHH looks odd because it looks regular, not because it is rarer than any other five-flip sequence: HTHHT and HHHHH each have probability 1/32.
Lottery draws work the same way. Which numbers came up in one round plays no part in the next. In the previous article, Hot and cold lottery numbers, 1,247 rounds of history showed that numbers absent for 20 or more rounds appeared in the next round 12.67% of the time, not different from the expected 13.33%. The data contain no phenomenon matching the feeling that a number is due.
"Such a long streak must be unusual" is also wrong
The twin of the gambler's fallacy is looking at a long streak and concluding "this is rigged" or "it must end now". In truth, streaks in random sequences are more common than intuition expects. The probability that fair-coin flips contain at least one run of the given length of heads:
| Flips | Run length | Probability |
|---|---|---|
| 20 | 5 or more | 25.0% |
| 100 | 5 or more | 81.0% |
| 100 | 7 or more | 31.7% |
| 100 | 10 or more | 4.4% |
| 1,000 | 10 or more | 38.5% |
In 100 flips a run of five heads appears with probability 81%. People who write down "random-looking" flips by hand tend to avoid long runs, so a genuinely random sequence often looks artificial. Intuition expects randomness to be more even and alternating than it is.
The law of large numbers feeds the belief
The fallacy is plausible because it misapplies a true law. The law of large numbers says that with many trials the proportion of heads approaches 1/2. It does not say that an excess of heads is made up by tails. Suppose 100 flips gave 60 heads. Over the next 1,000 flips we expect 500 heads, so 560 of 1,100 flips are heads, a proportion of 50.9%. The 60–40 imbalance was never compensated, yet the proportion moved much closer to 50%. The gap remains; it is simply diluted by new trials. The coin does not remember the imbalance and correct it.
The psychologists Tversky and Kahneman called the expectation that even small samples must resemble their population a "belief in the law of small numbers" (1971). We checked only that paper's bibliographic details, so we do not report its specific experiments.
The error appears in the real world
It is not confined to the laboratory. Clotfelter and Cook (1993) analysed Maryland's numbers game and reported that the amount wagered on a number fell right after that number won, then recovered over several months, which they read as consistent with the gambler's fallacy. Chen, Moskowitz and Shue (2016, Quarterly Journal of Economics) reported that decisions by asylum judges, loan officers and baseball umpires lean against the previous decision. For both studies we checked bibliographic records and summaries but not the full text, so we report only their headline claims.
How to think about it
- Each draw is independent. Earlier results cannot tell you that a number is "due" or "not due". The number generator on this site does not use earlier results either.
- Long streaks are common in random data. Before suspecting manipulation from a streak's length, work out how rare that length is under randomness.
- Size of the probability and "turn" are different things. A single ticket's 1 in 8,145,060 does not grow because you have played for years. More tries raise the chance of at least one success; each try keeps its own probability.
One limit: this page treats ideal independent trials. If a device is biased, or earlier results physically change later ones (for instance drawing cards without reshuffling), trials are not independent and probabilities change. Lottery draw machines are set up afresh for each draw, so this site assumes independence, and the 1,247 rounds above are consistent with that assumption.
Playing the same numbers versus changing them weekly
A common follow-up to the gambler's fallacy is: "Is it better to keep the same numbers or change them every time?" Mathematically the answer has two parts.
- Across different rounds there is no difference. Each purchase has a 1/8,145,060 chance of the jackpot and rounds are independent. Playing the same numbers for 10 weeks or different numbers each week gives the same chance of at least one jackpot in 10 weeks: 1 − (1 − 1/8,145,060)10, about 0.000123%. The thought that "numbers just drawn will stay away" or that "numbers that have not come are due" favours neither choice.
- Within one round there is a difference. Buying the same numbers five times in one round leaves the chance at 1/8,145,060. Five different tickets give 5/8,145,060. The chance rises in proportion to the number of distinct tickets, and the expected return is still negative at an average of KRW 500 per ticket. Duplicate purchases of one combination add no chance at all while costing money.
So the accurate answer to "should I change my numbers?" is: it makes no difference between rounds, and within a round non-overlapping tickets are better than overlapping ones. This is not a way to predict a win; it is the arithmetic of buying different combinations with the same money.
Practice questions
Three questions to check intuition. Answers follow.
- A coin has landed heads 6 times in a row. What is the probability the seventh flip is heads?
- What is the probability that 7 flips are all heads?
- In Lotto, a number has not appeared in the last 30 rounds. What is the chance it appears in this round?
Answers: (1) 1/2. The first six flips have already happened. (2) 1/128, about 0.78%. The difference between questions 1 and 2 is whether you are told that six flips have already occurred. Given that, the probability returns to 1/2; the 1/128 in question 2 applies before any flip has happened. (3) 6/45 = 13.33%. Having missed 30 rounds has no effect.
LottoFlick is independent and not an official lottery operator. Numbers are random and cannot be predicted.
Sources and verification
- Clotfelter & Cook (1993), "The 'Gambler's Fallacy' in Lottery Play", Management Science 39(12): 1521–1525 — partly verified · bibliographic details and main finding checked via library records; full text not read
- NBER Digest on Chen, Moskowitz & Shue (2016), Quarterly Journal of Economics 131(3) — partly verified · summary read; the paper's full text was not
- Tversky & Kahneman (1971), "Belief in the law of small numbers", Psychological Bulletin 76(2): 105–110 — partly verified · bibliographic details only
- LottoFlick aggregate output: next-draw rates by rounds since last appearance — computed by LottoFlick