“It can’t come up again.”
It came up again.

One distribution, three questions

Everything on this page is the same formula asked three times.

Take a draw of $k$ numbers from a pool of $N$. Now take another draw, made by a machine with no memory of the first. How many numbers do they share?

The second draw is a uniform $k$-subset, and the first draw's $k$ numbers are a fixed target inside the pool. That is a hypergeometric setup — drawing without replacement from a population split into "was in the last draw" and "wasn't":

$$P(X = j) = \frac{\binom{k}{j}\binom{N-k}{\,k-j\,}}{\binom{N}{k}}$$

The mean simplifies to something worth remembering:

$$\mathbb{E}[X] = k \cdot \frac{k}{N} = \frac{k^2}{N}$$

Each of your $k$ numbers has probability $k/N$ of being among the next draw's $k$, and expectation adds regardless of the dependence between them. For a 6/49 that is $36/49 \approx 0.73$ numbers carried over per draw.

Why "last week's numbers are used up" is backwards

The quantity players actually care about is $P(X \ge 1)$, and it is the complement of a single term:

$$P(X \ge 1) = 1 - \frac{\binom{N-k}{k}}{\binom{N}{k}}$$

For a 6/49: $1 - \binom{43}{6}/\binom{49}{6} = 1 - 6{,}096{,}454/13{,}983{,}816 \approx 0.564$.

More often than not, a 6/49 draw repeats a number from the draw before it. For a 7-from-52 game it is about 66%; for a 5-from-69 game, where the pool is sparse relative to the pick, it drops to 32%. The belief that a number is "used up" for a while has the arithmetic exactly inverted, and the strength of the effect is a property of the format rather than of the balls.

The same formula, over the whole history

Nothing in that derivation used the fact that the two draws were consecutive. The machine has no memory of a week ago and none of a decade ago either, so the same distribution describes every pair of draws in the history.

With $D$ draws there are $\binom{D}{2}$ pairs, and the expected number sharing exactly $j$ numbers is $\binom{D}{2} \cdot P(X = j)$. That is the ladder on this page — and its top rung, , is two draws that were completely identical.

The birthday problem, wearing a lottery's clothes

Here is where intuition fails hardest.

Asked whether a lottery has ever repeated a draw, people reach for $1/\binom{N}{k}$ — one in 13,983,816 for a 6/49 — and conclude it could not have happened. But that is the answer to "will the next draw match this specific one?" The real question is whether any of the $\binom{D}{2}$ pairs match, and that count grows quadratically while the space stays fixed.

The expected number of repeats after $D$ draws is

$$\mathbb{E}[\text{repeats}] = \frac{\binom{D}{2}}{\binom{N}{k}}$$

and, treating them as approximately Poisson,

$$P(\text{at least one repeat}) \approx 1 - \exp\!\left(-\frac{D(D-1)}{2\binom{N}{k}}\right)$$

Even odds arrive when $D(D-1)/2 \approx \binom{N}{k}\ln 2$, which gives

$$D = \left\lceil \frac{1 + \sqrt{1 + 8\ln 2 \binom{N}{k}}}{2} \right\rceil$$

For a 6/49 that is 4,404 draws — around forty years of twice-weekly play. Not the geological timescale the one-in-fourteen-million figure suggests. A game that has been running since the 1980s is not unlikely to have repeated itself; it is roughly a coin flip.

The correction nobody mentions

There is one more step, and this page learned it the hard way.

When you hunt for coincidences across many games at once, the expected count rises with every game you add — but so does the number of ways your data can be wrong. Scanning every lottery in this database for repeated draws returns about a third more than chance allows, and essentially none of the excess is remarkable: it is one physical draw recorded under two slot names, or a result posted to two consecutive dates.

The statistical lesson generalises well beyond lotteries. At the scale where coincidences become expected, data errors become expected faster, because they are not competing against a one-in-fourteen-million denominator. Any anomaly hunt that does not model its own error rate will find its bugs first and report them as discoveries.

Sofia, September 2009

On 6 September 2009 the Bulgarian national lottery drew 4, 15, 23, 24, 35 and 42. Nobody won.

Four days later, on 10 September, the machine produced 4, 15, 23, 24, 35 and 42 again. This time eighteen people had the ticket.

The reaction was immediate and entirely predictable. The country's sports minister ordered an investigation; commentators explained that the odds were millions to one and that such a thing could not happen by accident. The investigation found no evidence of manipulation, and mathematicians pointed out what the arithmetic on this page says: across the thousands of draws held every week by hundreds of lotteries worldwide, a repeat somewhere is not merely possible but expected.

The Bulgarian case is famous because it was consecutive, which is genuinely rarer than a repeat separated by years. But the instinct it provoked — a coincidence this striking must have a cause — is the one this whole playground exists to examine.

The two beliefs that cancel out

Lottery players hold two convictions about repetition, and they contradict each other.

The first is that a number which came up last week is spent, and should be avoided. This is the gambler's fallacy in its most common costume, and the arithmetic is not close: in a 6/49 game, more than half of all draws repeat a number from the draw before. The "unlikely" thing is the one that happens most weeks.

The second is that a whole draw repeating would be proof of a rigged machine. That one is wrong in the other direction — a repeat is exactly what a fair machine produces once a game has run long enough, and the point at which it becomes likely is far nearer than anybody guesses.

Both beliefs come from the same place: a sense that the machine ought to remember what it has already done. It doesn't. It has no state at all between draws, which is what makes both the carry-over and the eventual repeat inevitable.

The coincidence audit

Building this page produced a small object lesson, and it seemed dishonest to leave it out.

The first version simply hunted for repeated draws across every game in this database. It found 10,400 of them, against roughly 1,455 that chance allows — a sevenfold excess that, taken at face value, would be the largest anomaly ever reported in a lottery dataset.

It was not an anomaly. It was three separate bugs.

Some games run several sub-draws under one name — South Africa's Lotto has a main draw, Plus 1, Plus 2 and a 5 Max — and the same physical result was filed under two of those slot names, so it looked like the machine had repeated itself within a second. Another game had a stretch in 2024 where results were posted to two consecutive dates, producing sixteen "one in 177,100" events in a single spring. And the search itself had a flaw: it packed each draw into a 64-bit mask, which silently dropped every number above 64, making Powerball's 65 to 69 invisible and manufacturing a clean, plausible, entirely fictional deviation.

Only the last of those was our own code, and it was the most dangerous, because it did not look like a bug. It looked like a finding.

What that has to do with you

Every claim on this site is an anomaly hunt of some kind: is this number hot, is this pair unusual, is this machine fair. The honest version of that work spends most of its effort on the boring question — could this be an artifact? — because at the scale where genuine coincidences start appearing, mistakes are appearing much faster.

The register on this page is what survived that question. It shows the repeats we believe, the number chance predicted, and the ones we threw out and why. It is a shorter list than the first version produced. It is also true.