In Israel, to win the weekly Lotto lottery one has to correctly guess 6 numbers out of 37, and in addition to guess one number out of 8. On September 21 the numbers that came out were 13, 14, 26, 32, 33, 36, and the seventh number was 2. On October 16, very surprisingly, the very same 6 numbers were chosen, and the seventh number was 8. The country was left open mouthed. How can such a thing happen? Within one month, the same six numbers were chosen. A statistics consultant to the lottery organizer said that this event is statistically very rare. Nobody thought to doubt this statement. But I was not convinced.

Let’s do the calculations. One has to choose 6 numbers out of 37. The number of ways to do that is 37!/(31!*6!), which turns out to be 2,324,784. This means that if we have asked on October 15 “what is the probability that the six numbers that will come out tomorrow will be identical to those that came out on September 21″, the answer would have been about 1 over two million. But this is not what happened. We already know the results, so in fact our question should be “what is the probability to be surprised by the Lotto lottery?” This is a tricky question: we would be surprised if the six numbers that come out are the same as the six numbers that came out last week, or two weeks ago, or 6 weeks ago. We would be surprised if the 6 numbers turn out to be 1,2,3,4,5,6, or 2,4,6,8,10,12. And the surprise could have happened last week, or two weeks ago. That is, if the same six numbers were chosen on February 2, 2008, and on February 16, 2008, we would have been surprised in a similar way on February 16, 2008, instead of being surprised this week.

To simplify matters, let’s calculate the probability of the event “the same six numbers come out twice in nearby dates in all lotteries made so far”. The website of the Israeli Lotto lists the outcome of the last 731 lotteries; a simple calculation shows that the probability that the same six numbers come out in two lotteries within the same two months in a period including 731 lotteries is roughly 1/400. This is a small number, but not “statistically rare”.

Fine. So statistical consultants do not know which probability to calculate. This is disappointing but not too surprising. To rectify the statements that were made in the news I talked with the Science journalist of one daily newspaper, I wrote a short OpEd explaining this point and sent it to the daily newspaper and to an internet news portal. Nothing. They did find this point sufficiently important, not even to devote for it a virtual space in their website. That was more surprising to me. To publish incorrect yet seemingly sound statements about the rarity of a certain event is fine, but to publish a surprising statement that the occurrence of the event is in fact not surprising is too much. This was a real disappointment. The only tool left for me was this blog. I decided to use it.

## 11 comments

October 21, 2010 at 5:04 pm

AssafFurthermore, on the main article on that in a big Israeli portal they even went further and claimed that the probability of having two lotteries with the same 6 numbers is in fact (1/2M)^2…

Here:

http://www.ynet.co.il/articles/0,7340,L-3970398,00.html

October 22, 2010 at 1:35 am

randoi’d call 1/400 “statistically rare”.

October 24, 2010 at 7:55 pm

apartamentosi’d call 1/400

October 25, 2010 at 3:26 pm

Instalaciones Deportivasprobabilities are kinda slick. I never trust them

October 26, 2010 at 1:52 pm

anonymousCorrection:

To publish incorrect yet seemingly sound statements about the rarity of a certain event

sells papers, but to publish a surprising statement that the occurrence of the event is in fact not surprisingdoes not.November 2, 2010 at 2:35 pm

riseaThere are 37 days in a year…(and 8 other strange nights)… What is the probability that of 731 families of six , two of the cohorts will have the same birthdates…

November 8, 2010 at 4:33 pm

OneWinner.MeAstonishing result indeed if it is correct.

Will try to find the calculus that leads to the 1/400 result …

November 13, 2010 at 12:06 am

Chris AngellPerhaps some people believe that the balls exited the machine in the same order for both lotteries. In your math, order does not matter. I bet some people think order matters.

November 13, 2010 at 10:26 am

EilonThe lottery is broadcasted real-time on TV, so you can in fact verify that everything was fair. And the order in which the numbers were chosen in the two lotteries was different. The only conclusions that I can reach is that (a)people asked the wrong question, and (b) as Assaf mentioned above, they calculated the incorrect answer to the question that they asked.

December 7, 2010 at 7:52 am

2thoughts1. They are surprised because they are Bayesian. There are 2 states of the world: one state the machine is fair and draws are IID. The other state is less likely but not zero, that machine is rigged or balls are not properly mixed etc.

They are surprised that their belief in the second state just shot up.

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2. Israel is only special to a select group for the surprising context (not me). To outside observer, I could’ve been reading about France, USA, local, state, national lottery etc. What is the probability that something like that occurred? Higher than 1/400. What’s the prob. given occurred that it’s picked by news and re-circulated? Approaches 1.

March 9, 2011 at 3:03 pm

Loto israélien : un phénomène si exceptionnel que cela ? | OneWinner.Me – Le blog[...] Et vu sous cet angle, les calculs montrent que la probabilité de l’évènement n’est pas aussi minime que ce que les premiers commentaires hâtifs ont pu nous faire penser ! Le phénomène a même de forte chances de se produire (1 chance sur quelques centaines à quelques milliers suivant les calculs, c’est à dire une apparition en quelques années). C’est ce que montrent plusieurs raisonnements comme celui-ci ou celui-là. [...]