Last week, the Freakonomics blog in the NYT reported that the Israeli lottery had drawn the same six numbers twice in a month. The seventh "bonus ball" was different, but still: quite a coincidence, right? Cue the quote from an expert to explain just how remarkable this is:

Yitzhak Melechson, a statistics professor at the University of Tel Aviv, said that “the incident of six numbers repeating themselves within a month is an event of once in 10,000 years.”

Hmm, not so much. As Xi'an pointed out, there's actually about an 8% 8 in 1000 chance of this having occurred in the two years the Israeli lottery has been run under these rules. And then if you consider all the lotteries around the world, as we did when the Bulgarian lottery had back-to-back identical drawings just over a year ago, then this isn't looking so surprising after all. It's likely that Yitzhak Melechson was misquoted, but still, as Gelman says, it's surprising the guys at Freakonomics missed this one.

**Update**: Corrected estimate of the probability of coincidence. Apologies for the error, which was mine, not Xi'an's.

Freakonomics blog: Very Long Odds in the Israeli Lottery

Xi'an didn't say 8 percent, he said 8 per mil. It's a once in a century event for that lottery, not a once every dozen years.

Posted by: djm | November 02, 2010 at 14:38

Thanks djm. My error: I read, "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.", but missed it was a ‰ symbol, not a % symbol. Thanks for pointing out my error, which given the topic of this post, was an embarrassing one!

Posted by: David Smith | November 02, 2010 at 15:50

a> the incident of six numbers repeating themselves within a month is an event of once in 10,000 years

b> there's actually about an 8 in 1000 chance of this having occurred in the two years the Israeli lottery has been run under these rules.

How does (b) contradict (a)? If I understand (b) correctly, there is a 8 in 1000 chance of "six numbers repeating themselves" in a twenty-months period (since the creation of the lottery), but it says nothing about the probability of an event of the type "six numbers repeating themselves _within_a_month_".

Posted by: carlitos | November 02, 2010 at 18:03

Actually I do not see any contradiction,

see

http://freakonometrics.blog.free.fr/index.php?post/2010/11/02/Comments-on-probabilities

Posted by: Arthur | November 03, 2010 at 12:41

Apologies to everyone for using the "\permil" symbol that confused readers.... I also added a computation to my post to answer a question similar to carlito's.

Posted by: xi'an | November 03, 2010 at 12:54