Israel’s national lottery is all over the news these days because the same numbers (13, 14, 26, 32, 33, and 36) came up twice during one month.

A journalist called an expert on gambling, Z. Gilula, a professor of statistics in Israel, and asked him about the probability of the same set of numbers being randomly picked twice.

They draw 6 numbers out of 37, thus there are 37!/(31! 6!) = 2324784 different possible drawings. Therefore, the probability of any one set of six numbers is p = 1/2324784 = 4.30 * 10^-7. To get this twice in a row, you need to square it, i.e. the probability is really extremely small: p² = 1.85 * 10^-13

And now let us ask the questions that should have been asked and which mister, sorry, professor Gilula should have been aware are actually the proper questions to answer:

Given two drawings a week, how big is the probability that the same set of 6 numbers appears twice or more in any one month (that is 8 consecutive drawings) in any one particular such lottery (like the Italian, the German national lottery, ...), considering that there are many months (maybe 50 odd years of history of national lotteries) and moreover taking into account that there are many such lotteries all over the world (maybe 100 comparable ones counting the many countries that have them)?

The answer is: The probability is basically equal to 1, that is, this “wonder” was bound to happen. Don’t believe it? Let us start with the easier part: Given two drawings a week, how big is the probability that the same numbers appear next month in that particular Israeli lottery?

Say that the next weekend’s combination is D₁ (read: drawing number one, which is a set of six numbers) and call the seven drawings after that D₂ to D₈. The drawings are independent, and so the probability for D_i = D_j with i≠j is the above p every time. The probability of no such coincidence D_i = D_j with i≠j equals (1 - p)^7+6+5+…+1.

That means that the total probability for no such coincidence is 1 - P = (1 - p)²⁸ or in turn that the probability for such a coincidence is P = 1 - (1 - 31! 6! / 37! )²⁸ = 1.20 * 10^-5.

Yep, you see this correctly, P is almost 30 times higher than the probability of your winning the lottery (which is p)!

A coincidence over three months would probably still make the news, and in that case one gets:

P = 1 - (1 - 31! 6! / 37! )²⁷⁶ = 1.19 * 10^-4.

A very rough estimate would be that there are 200 three month periods in 50 years, although that would underestimate the probability. Anyways, add the many different lotteries that are played all over the world, and you can certainly appreciate that this was all bound to happen at some point.

What do we learn? Do not play the lottery, send the money for the ticket directly to me.