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Lottery Birthday Paradox: Why Draws Repeat

The lottery birthday paradox explains why the same lottery numbers drawn twice is less rare than it feels. The calculator above shows the repeat chance for any number of draws.

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Woman surprised by two matching balls

Why repeats come early

  1. Count pairs, not drawsA repeat needs any two draws to match. Every new draw is compared with every earlier draw, not only with the last one.
  2. Pairs grow fastEach extra draw adds a pair with every earlier draw, so the number of pairs grows much faster than the number of draws.
  3. Same as birthdaysIn a group, a shared birthday appears sooner than people expect for the same reason: many pairs, each with a small chance.
  4. One set is still rareOne specific earlier draw coming back is very unlikely. Some match among all the pairs is what becomes likely.

Chance of a repeat

P(repeat) = 1 − product of (1 − i ÷ C) for i = 1 … n − 1; C = possible draws, n = draws

Example: For example, with C = 133,784,560 Lotto Max sets and n = 5,000 draws, P(repeat) = 8.9%.

Draws until a repeat is likely

Draws for an even chance

13,619 draws

After this many Lotto Max draws, a repeat of some full draw is more likely than not.

Time at Lotto Max pace

131 years

Lotto Max draws take place every Tuesday and Friday, so reaching that number of draws takes about this long.

Example with fewer draws

8.9%

For example, after 5,000 draws, this is the chance that some full draw has already repeated.

One set vs any two draws

One chosen set repeats

  • You fix one set of numbers in advance
  • Each draw is one of 133,784,560 equally likely sets
  • Its chance stays the same in every draw
  • Past draws do not change it

Any two draws match

  • Every pair of draws can match
  • Pairs add up quickly as draws pile up
  • After 5,000 draws the chance is 8.9%
  • After 13,619 draws a repeat is more likely than not

Words on this page

Full draw
The seven main numbers of one Lotto Max draw. The bonus number is not counted here.
Repeat
Two different draws with exactly the same set of main numbers, in any order.
Possible draws (C)
How many different sets the game can produce. For Lotto Max main numbers it is 133,784,560, counted as combinations.
Birthday paradox
The fact that some match among many items appears much sooner than a match with one chosen item.

Sources

Questions

Can the same lottery numbers be drawn twice?

Yes. Each Lotto Max draw is one of 133,784,560 possible sets of main numbers, and earlier draws do not block any set. After 5,000 draws, the chance that some full draw has repeated is 8.9%.

What is the birthday paradox?

In a group, a shared birthday shows up sooner than most people guess, because every pair of people is a chance for a match. The same counting applies to pairs of lottery draws.

Does a repeat draw mean the lottery is rigged?

No. Random draws produce repeats at the rate the formula predicts. A repeat also says nothing about the next draw, because every draw is independent of the ones before it.

How many draws until a Lotto Max draw is likely to repeat?

About 13,619 draws. With draws every Tuesday and Friday, that takes about 131 years.

Does a past repeat make some numbers more or less likely?

No. Every combination has the same chance in every draw, whether it was drawn before or not, and quick pick and own numbers have the same odds.