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

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

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

,

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

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

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%!