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Probability illusions: the hot hand, ignoring size and vivid wins

The same number can be read very differently by different people. Shown "1 in 8,145,060", one person says "someone does win, though" and another behaves as if ten tries would surely do it. Knowing a probability in your head and feeling it are different things, and the gap has a few recurring patterns. This page looks at the three closest to lotteries: the hot hand, ignoring the size of a probability and the availability heuristic. We start with the illusion that sits opposite the "it's due" belief from the gambler's fallacy.

1. The hot hand: a streak will go on

The gambler's fallacy says a streak must end. The hot hand says it will continue. In basketball a player who has made several shots in a row is said to have a "hot hand". Gilovich, Vallone and Tversky (1985) analysed shooting records of the Philadelphia 76ers, free-throw records of the Boston Celtics and a controlled experiment with Cornell varsity players, and reported no positive correlation between successive shots. Previous results shaped players' predictions of their next shot but not their actual success. The authors explained the belief as an expectation that even short random sequences should resemble the process generating them.

That conclusion has since been contested. Miller and Sanjurjo (2018, Econometrica) argued that the standard way of measuring success after a streak has a selection bias that can hide a small effect in short sequences, and that correcting it reverses earlier conclusions. We confirmed only the abstracts of both papers, so we do not judge who is right. What matters here is this: the debate is about human skill. People's form rises and falls, so a domain may exist where streaks mean something. A lottery draw, by contrast, is a machine process with independent draws. It has no form and no skill. A hot hand in lottery numbers, "recently frequent numbers keep coming", has no structure to rest on. In 1,247 rounds, numbers that had appeared three or more times in the previous ten came up next at 13.62%, equal to the expected 13.33% (see the hot-number article).

That one person looks at the same run and says "it must end" and another says "it will continue" shows that people look for meaning first when reading chance. In independent draws neither intuition has a basis.

2. Ignoring size: small chances all look equally small

The second illusion is failing to feel differences in size among very small probabilities. 1 in a million and 1 in 8 million differ eightfold, yet both blur into "almost never". So behaviour does not change when probability does. Intuition's scale barely distinguishes within the very small range.

To make the gap felt, restate it. If one game wins the jackpot with probability 1 in 8,145,060, how many games are needed for a 50% chance of at least one win?

Games needed for a given chance of at least one jackpot (1 in 8,145,060 per game)
Target chanceGames neededBuying 5 games (KRW 5,000) a week
50%5,645,726about 21,700 years
90%18,754,693about 72,100 years

Buying five games a week for 100 years is 26,000 games, giving a 0.32% chance of at least one jackpot. Change the probability and count in the tool below. Entering 0.0000123 (%) in the per-try box matches Lotto 6/45 almost exactly.

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Another way to feel the size is to look at how often the event actually occurred. In rounds 88–1247, which this site analysed, roughly 84.7 billion games were sold and there were 10,387 first prizes, about one per 8.15 million games. It takes that volume of sales for about nine winners to appear each week. "Someone will win" is true because tens of millions of games are pooled, not because anyone's single ticket becomes likelier.

3. The availability heuristic: what comes to mind is common

The third habit is judging probability not by calculation but by how easily examples come to mind. Tversky and Kahneman (1973) reported that people judge the frequency of a class or the probability of an event by how readily instances come to mind, and that this yields systematic biases. We confirmed only the bibliographic record of that paper.

In lotteries the bias works strongly. A jackpot becomes news, winners give interviews, stores hang banners. The tens of millions of losing tickets are recorded nowhere. Only memorable cases are visible, so winning feels more common than it is. An unusual round like 1128, with 63 winners, leaves an even stronger impression. Yet in that same round more than 111 million games lost.

Availability works in the other direction too. If numbers you have bought now and then never came up, the sense that "next time they will" is not a judgement about probability but a feeling about expectation.

When the three combine

The illusions often work together. A few repeats of recent numbers read as a hot hand, numbers that have not come read as "due", and all of that rests on the memory that "a first prize can happen to an ordinary person", which makes one forget the size of the probability. The result is a feeling that "I have bought so many, I must be close". Nothing has come closer. Each game is independent, so the number bought before does not change the next game's probability.

Ways to calibrate intuition

Questions to try yourself

Four questions to see how far the three illusions enter your judgement. Answers follow.

  1. In Lotto, is the chance that a number which appeared three or more times in the previous 10 rounds comes up this round higher than the base rate of 13.33%?
  2. A game succeeds with probability 1% per try. Is the chance of at least one success in 100 tries 100%?
  3. One lottery has a first-prize chance of 1 in a million, another 1 in 8 million. If the prize is the same, how many times different is the expected value?
  4. You saw this week's jackpot winner interviewed on the news. What does that tell you about your own chance?

Answers: (1) No. In the 1,247-round record such numbers came up next at 13.62%, at the level of the 13.33% base rate. (2) No. 1 − 0.99100 ≈ 63.4%; even after 100 tries 36.6% of people have still not succeeded. (3) Eight times: with equal prizes the expected contribution differs eightfold. (4) Nothing. An interview is one winning case and says nothing about the tens of millions of games that did not win.

Many people feel that 100 tries at 1% means 100%. That an expected count of one success in 100 tries does not mean certainty is close to ignoring size. Enter probability 1 and tries 100 in the tool above and it shows 63.4%.

Limits

These illusions are not a matter of intelligence or education; researchers fall for them too. So the remedy is not to blame intuition but to reach for a calculator where intuition tends to fail. The next article, Why big numbers blur our judgement, continues the theme.

LottoFlick is independent and not an official lottery operator. Numbers are random and cannot be predicted.

Sources and verification