Hot and cold numbers, honestly

What every number should have done

Euro Millions Plus has drawn 206 times in this rule era. Each draw takes 5 numbers from a pool of 50, so any particular number is in a draw with probability $p = k/N$, and across $D$ draws it should turn up:

$$E = D \cdot \frac{k}{N}$$

which here is about 20.6 appearances per number. That is the line every stem hangs from. Nothing on the chart is "above average" in an interesting sense — half of them have to be.

How far from that line is normal

An appearance is a coin flip with probability $p$, repeated $D$ times, so counts follow a binomial distribution:

$$\sigma = \sqrt{D \cdot p \cdot (1-p)}$$

For this era that is 4.3. The shaded band is ±2σ, and about 95% of the numbers should sit inside it — not because the machine is being careful, but because that is the shape randomness has.

Here is every number's z-score, $z = (O - E)/\sigma$, against the bell curve they should follow if nothing is going on:

The bars are the lottery. The curve is chance. That is the whole argument.

The test a regulator would run

Rather than squinting at individual numbers, add up all the deviations at once:

$$\chi^2 = \sum_{i=1}^{N} \frac{(O_i - E_i)^2}{E_i}$$

For this era, χ² = 63.1 on 49 degrees of freedom, giving p = 0.085. Consistent with a fair machine. The variation between numbers is exactly the size randomness produces.

One wrinkle, and it matters. The numbers in a single draw are not independent: a draw takes $k$ balls without replacement, so if 7 comes up, something else did not. That negative dependence shrinks the raw statistic, and using it uncorrected makes every lottery look tidier than it is — a bias in the flattering direction. Rescaling by $(N-1)/(N-k)$ puts it back on the scale it is read against. This is the practical form of the modified statistic Joe (1993) derived for exactly this situation.

Why none of it predicts the next draw

The balls have no memory. They are not aware of the chart, of their own history, or of how long it has been. Formally, draws are independent events, so for any number:

$$P(\text{drawn next}) = \frac{k}{N}$$

regardless of whether it came up last week or has been missing for a year. A number that is "due" is a number that has been unlucky, and unluckiness is not a force that corrects itself.

The mirror-image error is just as common: believing a hot number is on a run and will keep going. Both mistakes assume the past leans on the future. It does not.

The gambler's fallacy, measured in cash

In 1993 Charles Clotfelter and Philip Cook went looking for the fallacy in real money. Maryland ran a daily numbers game where players pick a three-digit combination, and the state kept records of exactly how much was bet on each one.

What they found was stark. The day after a number won, the money staked on it collapsed — players avoided it, certain it could not come up again so soon. The amount recovered slowly, taking several months to return to normal. The belief was not vague; it was worth measurable sums, and it cost people real money for months at a time.

Donald Terrell replicated it a year later in a pari-mutuel game, where picking an unpopular number genuinely pays better because you share the prize with fewer people. The effect was still there — but weaker. People held the superstition somewhat less tightly when it had a price tag attached.

The audit is the chart you are looking at

The χ² test in the maths panel is not a toy. It is the standard tool for answering "is this machine fair?", and it has been pointed at national lotteries repeatedly.

Hugh Joe worked out the corrected form for lotto draws in 1993. John Haigh applied the machinery to the UK National Lottery in 1997, and the University of Salford ran a formal randomness analysis of it in 2004–05. The verdict, every time: no significant departure from randomness.

That is a boring headline and an important one. When a statistician tests a lottery and finds nothing, the finding is the story.

When the chart would have screamed

Deviation-hunting is not useless — it is how the fraud gets caught.

On 24 April 1980, the Pennsylvania Daily Number drew 666. It should have been a 1-in-1000 result. It was not: the balls had been injected with latex paint so that only the 4s and 6s were light enough to rise, leaving eight possible outcomes instead of a thousand. The scheme was exposed not by the draw itself but by the betting patterns — a flood of money on a handful of combinations, which is a deviation of exactly the kind this page is built to detect.

A weighted-ball rig would light this chart up like a bonfire. That it stays flat, year after year, is the honest result.

The one thing that actually helps

Number history will not improve your chances — nothing can, short of buying more tickets. But number popularity can improve your payout, because prizes are shared and most people pick badly. That is a real edge, it is small, and it is the only one on offer.