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

\choose{37}{6}\times 10=</span></span></span>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

18 Responses to “Coincidence in lotteries”

  1. [...] Xi’an did (here), assume now that there is a lottery over 100 countries. Here I define “coincidence” [...]

  2. [...] ask about the order the numbers came out…) Xi’an published also a post on that topic, there, since last week, the same thing happened in Israel. All that reminded me a discussion I had with a [...]

  3. [...] I found a site entitled Mr Meyrowitz’s Class that links to my first post on coincidences in lotteries as an example of “fatal error”. This seems to be part of a [...]

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