Foundational heuristics-and-biases research discusses how representativeness can distort judgments about sequences. The worked examples below distinguish that error from a real change in probability or dependence between events.
Fixed independent example
Assume, for illustration, a fair coin with independent flips. The probability of heads on any one flip is 0.5. After five tails, the probability of heads on flip six remains 0.5. The past sequence has not changed the coin or the next flip.
The probability of observing exactly the ordered sequence T-T-T-T-T-H from the start is:
0.5^6 = 0.015625, or 1.5625%.
That small probability for the complete pre-specified sequence is not the conditional probability of heads after five tails. Under independence, P(H on flip 6 | first five are tails) = 0.5. Independent events require that knowing one event does not change the probability of the other.
When sequence history can matter
| Situation | Question | Why it differs from the fallacy example |
|---|---|---|
| Team matches | Have players, tactics, opponents, venue, or schedule changed? | Probabilities can vary between events |
| Repeated model errors | Is there evidence of drift or a broken input? | The forecasting process may have changed |
| Market execution | Have price, limits, or settlement terms changed? | The contract is not identical |
| Physical device | Is there evidence the mechanism is biased or changing? | Trials may not share a fixed probability |
| Related outcomes | Does one event alter the next state? | Independence may fail |
Real football results are not repeated fair-coin flips. Opponents, lineups, venues, and prices vary. That does not make a reversal due; it means the analyst needs a model of the changed process.
Do not confuse it with regression to the mean
Regression to the mean can occur when unusually extreme measurements are followed by measurements closer to a longer-run level under specified repeated-measurement conditions. It is not a law that the opposite result must happen next. Use the dedicated regression-to-the-mean guide for that distinction.
Sequence audit
- Define the next event and target.
- State whether trials are assumed independent.
- Identify every reason the underlying probability might change.
- Estimate from current information, not a due-outcome narrative.
- Compare the forecast with a baseline on later events.
- Record uncertainty and avoid a bet when the process is not identified.
Continue learning
- Next guide: Illusion of Control in Betting
- Related guide: Loss Aversion in Betting
Assumptions and limitations
The coin is a teaching model, not a football forecast. Independence and a fixed 0.5 probability are assumptions. Real sequences can contain dependence, changing rates, selection effects, and measurement error. Rejecting the gambler's fallacy does not make a separate probability estimate accurate or a price favourable.
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