## Coincidence in lotteries

Last weekend, my friend and coauthor Jean-Michel Marin was interviewed (as Jean-Claude Marin, sic!) by a national radio about the probability of the replication of a draw on the Israeli Lottery. Twice the same series of numbers appeared within a month. This lotery operates on a principle of 6/37 + 1/8: 6 numbers are drawn out of a pool of numbers from 1 to 37 and then an 7th number is drawn between 1 and 8. The number of possibilities is therefore

${37\choose 6}\times 8=18,598,272$

and the probability of replicating, on a given day, the draws from another given day is 1/18,598,272. Now, the event picked up by the radio does not have this probability, because the news selected this occurrence out of all the lottery draws across all countries, etc. If we only consider the Israeli Lottery, there are two draws per week, meaning that over a year the probability of no coincidence is

$\dfrac{18,598,272\times 18,598,271\times\cdots\times 18,598,168}{18,598,272^{104}}=0.9997$

namely that a coincidence occurs within one year for this particular lotery with probability 3/10,000. If we start from the early 2009 when this formula of the lotery was started, there are about 188 draws and the probability of no coincidence goes down to

$\dfrac{18,598,272\times 18,598,271\times\cdots\times 18,598,084}{18,598,272^{188}}=0.999$

which means there is more than a 1‰ chance of seeing twice the same outcome. Not that unlikely despite some contradictory computations! It further appears that only the six digits were duplicated, which reduces the number of possibilities to

${37\choose 6}=2,324,784$

Over a month and eight draws, the probability of no coincidence is

$\dfrac{2,324,784\times 2,324,783\times\cdots\times 2,324,776}{2,324,784^{8}}=0.99999,$

which is indeed very small. However, if we start from the early 2009, the probability of no coincidence goes down to 0.992, which means there is close to an 8‰ chance of seeing twice the same outcome since the creation of this lottery… If we further consider that there are hundreds of similar lotteries across the World, the probability that this coincidence [of two identical draws over 188 draws] occurred in at least one out of 100 lotteries is 53%!

Last weekend, my friend and coauthor Jean-Michel Marin was interviewed (as Jean-Claude Marin, sic!) by a national radio about the probability of the replication of a draw on the Israeli Lotery. Twice the same series of numbers appeared within a month. This lotery operates on a principle of 6/37 + 1/8: 6 numbers are drawn out of a pool of numbers from 1 to 37 and then an 7th number is drawn between 1 and 8. The number of possibilities is therefore

$latex \choose{37}{6}\times 10=18,598,272$

and the probability of replicating, on a given day, the draws from another given day is 1/18,598,272. Now, the event picked up by the radio does not have this probability, because the news selected this occurrence out of all the lotery draws across all countries, etc. If we only consider the Israeli Lotery, there are two draws per week, meaning that over a year the probability of no coincidence is

$\dfrac{18,598,272\times 18,598,271\times\cdots\times 18,598,168}{18,598,272^{104}}=0.9997065$

namely that a coincidence occurs within one year for this particular lotery with probability 3/1000. If we start from the early 2009 when this formula of the lotery was started, there are 655 days and the

### 19 Responses to “Coincidence in lotteries”

1. According to Mr Meyrowitz, I made “a fatal error with decimals and probabilities”… According to his assignment, it can be found in less than 10 minutes,

• Is it [again] the same comment of readers missing the distinction between 1‰ and 1% [and 8‰ versus 8%]…?!

2. […] again, meaningless figures are published about a man who won the French lottery (Le Loto) for the second […]

3. […] (aka Jean-Claude!) Marin came for a few days so that we could make late progress on the revision of our book […]

4. […] Coincidence in lotteries – […]

5. Berwin Turlach Says:

Your calculation seems to allow the replicate draws to have any time difference between them. So you seem to address the question “What is the probability that I see at least one repeated draw over X years?”. Wouldn’t it be more relevant to ask “What is the probability of seeing a repeated draw within a month over X years?”.

• My goal in writing the post was primarily to address the absurdity of isolating these two identical draws from the myriad of other draws in order to conclude that there was nothing exceptional in this event when considering the correct universe. It is also possible to compute the probability of no repeated draw within one month with 8 draws over two years, as (modulo a possible flaw in my reasoning!)

$\dfrac{n(n-1)\cdots(n-7)(n-7)^{m-8}}{n^m}$

where n=2324784 is the number of combinations and m=192 the total number of draws. This leads to a probability of 0.0006 of seeing twice the same draw within a month. This is of course much smaller than the probability of seeing a repeated draw at all, however, the month horizon is somehow arbitrary and someone could as well find a draw within six weeks or two months as exciting and “impossible”…

6. I almost got caught by the percent/per-thousand thing too.

7. Ethan Brown Says:

If there is a 0.992 probability of no coincidence, wouldn’t that mean that there’s a 0.8% chance of a coincidence? You wrote 8%, and similarly wrote a 1% chance of a coincidence with a 0.999 probability of no coincidence. Or am I missing something?

• Ethan: this may be difficult to spot depending on your resolution/magnification but I did use the per mil (‰) rather than per cent (%) sign in both cases! Thanks for the careful reading!

8. Manoel Galdino Says:

Just a minor correction. 0.999 = 0.1% and 0.992 = 0.8%.

• Manoel: thanks for the careful reading, as in the reply to Ethan: I use the per thousand (‰), not the percent (%), sign! You are right too!

9. David Smith Says:

The same lottery draw occurring twice in a row has happened recently elsewhere, in Bulgaria. I took a look at the likelihood of that happening back then: http://blog.revolutionanalytics.com/2009/09/a-coincidence-occurred-film-at-11.html

• According to Jean-Michel, the reporter who interviewed him was not at all interested in the large scale picture you and I draw at explaining why such “impossible” events occur by de-conditioning over all draws and lotteries… ’tis presumably more exciting “news” to talk of a one in a billion event!

10. tu m’excuseras mais je ne retiens que le JEAN-CLAUDE!!!!

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