When the Same Numbers Came Up Twice

A lottery drawing the exact same six numbers it drew a few weeks earlier feels like proof that something is broken. It is one of the most reliable ways to get a national regulator to open an investigation, and it has happened enough times to be studied properly.

The investigations keep reaching the same conclusion, and the mathematics explains why.

The Israeli case, 2010

On Tuesday 21 September 2010, Israel's state lottery Mifal HaPayis drew 13, 14, 26, 32, 33, 36.

On Saturday 16 October 2010 — three and a half weeks later — the same six numbers came up again, this time in reverse draw order. The supplementary "strong number" differed: 1 in September, 2 in October (CNN, Haaretz).

Mifal HaPayis suspended the Saturday result and checked the machine for mechanical failure or tampering. Finding no irregularity, it certified the draw and paid out. Three players matched all seven numbers; a much larger group — reported at between roughly 92 and 125 depending on the outlet — matched the six main numbers, an unusually high count driven by players who had deliberately re-entered September's numbers (Irish Times).

Two statisticians were quoted at the time with wildly different figures. Yitzhak Melechson of Tel Aviv University called it "an event of once in 10,000 years"; Zvi Gilula of the Hebrew University put it near "one in four trillion" and compared it to the probability of life on Mars. They cannot both be right, and as the arithmetic below shows, neither is describing the question people were actually asking.

A note on dates: this event is frequently reported with the two dates transposed, as a 21 October draw repeating one from 16 September. The correct sequence is 21 September 2010 followed by 16 October 2010.

The Bulgarian case, 2009

A year earlier, Bulgaria's national lottery drew 4, 15, 23, 24, 35, 42 on 6 September 2009 and the identical six on 10 September 2009 — in consecutive draws, which is a considerably stronger coincidence. The government investigation found no manipulation. That case has its own article: the Bulgarian repeat draw.

Why a repeat is expected: the birthday problem

The instinctive calculation is the jackpot odds — 1 in 2,324,784 for a 6-from-37 game — and that calculation answers the wrong question. It gives the chance that a specific, named combination comes up. Nobody named 13-14-26-32-33-36 in advance. The actual question is whether any two draws in the whole history matched, and that is the birthday problem.

With M possible combinations and D draws, the number of pairs of draws is D(D − 1)/2, and the expected number of matching pairs is:

E[repeats] ≈ D(D − 1) / (2M) ≈ D² / (2M)

The draw count at which a repeat becomes more likely than not is:

D₅₀ ≈ √(2M × ln 2) = √(1.386 × M)

Israel's Lotto, 6 from 37

• M = C(37, 6) = 2,324,784
• D₅₀ = √(1.386 × 2,324,784) = √3,222,825 = 1,795 draws

At two draws a week — 104 a year, which is the schedule the September and October 2010 dates fall on — 1,795 draws is about 17 years. Extend that to 25 years of history, roughly 2,600 draws, and the expected number of repeats is:

2,600² / (2 × 2,324,784) = 6,760,000 / 4,649,568 = 1.45

More than one repeat expected. The 2010 event was not a coincidence that beat 4-trillion-to-one odds. It was the arrival of something the game was, by then, more likely than not to have already produced.

How the threshold varies by game

Divide the last column by the game's draws per year to convert to elapsed time. At 104 draws a year, Israel's threshold is 17 years and UK Lotto's is 76 years — which is exactly why a repeat has been recorded in the first and not the second.

The pattern is the birthday problem's signature: √M rather than M. Squaring the jackpot odds is the wrong instinct; taking their square root is the right one.

Consecutive repeats are genuinely much rarer

The Bulgarian case is a different animal. The probability that draw n + 1 exactly repeats draw n is 1/M, and over D draws there are only D − 1 consecutive pairs, not D²/2. Expected consecutive repeats:

E ≈ (D − 1) / M

For a 6-from-42 game with M = 5,245,786 across 5,000 draws, that is 0.00095 — about 1 in 1,050. Rare for one game.

But there are hundreds of six-number games running worldwide, most of them for decades. If a hundred such games each accumulate 5,000 draws, the expected count rises to roughly 0.1 — and across the far larger real population of games and their longer histories, "somewhere, at some point, a game repeated itself back to back" moves from surprising to eventually inevitable. This is why patterns are guaranteed restated in a different currency: rare events are certainties once you stop counting the opportunities.

What a repeat is not evidence of

It is not evidence the draw is rigged. A rigged draw is a predictable draw. Repeating a recent, widely publicised combination is close to the worst thing a manipulator could do, because it maximises the number of winners and therefore the payout — in Israel's case turning a single-winner jackpot into a hundred-odd claims. Real draw fraud looks like the Eddie Tipton hack or the 1980 Pennsylvania fix: a small set of combinations, bought quietly, claimed by associates.

It is not evidence the machine has a memory. The draw on 16 October had no access to the draw on 21 September. Each is an independent selection, and the probability of any particular combination was identical both times — see the gambler's fallacy.

It does not make those numbers "due" or "lucky". They carry exactly the same probability as every other combination, and after a publicised repeat far more people play them, which makes them a worse choice — not because they are less likely to win, but because you would split. See is 1-2-3-4-5-6 less likely? and the number sharing risk tool.

What would actually be suspicious

A repeat rate above the birthday-problem prediction, sustained across many draws. A repeat of a combination that had been heavily and unusually bought in advance. Or a physical finding — ball mass, machine wear, geometry — of the kind covered in ball weight and machine bias testing and what draw auditing involves.

A single repeat, in a game that has run long enough for √M draws, is the model working. The coincidence has been examined formally in the academic literature — Pollanen's 2024 paper A Double Birthday Paradox in the Study of Coincidences treats the Bulgarian case directly.

Simulate it for yourself with the draw simulator: run a 6-from-37 game for two thousand draws and watch a repeat show up.