Xian pointed me to this recycling of a classic probability error. It’s too bad it was in the New York Times, but at least it was in the Opinion Pages, so I guess that’s not so bad. And, on the plus side, several of the blog commenters got the point.

What I was wondering, though, was who was this “Yitzhak Melechson, a statistics professor at the University of Tel Aviv”? This is such a standard problem, I’m surprised to find a statistics professor making this mistake. I was curious what his area of research is and where he was trained.

I started by googling Yitzhak Melechson but all I could find was this news story, over and over and over and over again. Then I found Tel Aviv University and navigated to its statistics department but couldn’t find any Melechson in the faculty list. Next stop: entering Melechson in the search engine at the Tel Aviv University website. It came up blank.

One last try: I entered the Yitzhak Melechson into Google Scholar. Here’s what came up:

Your search – Yitzhak Melechson – did not match any articles

Computing wrong probabilities for the lottery must be a full-time job! Get this guy on the Bible Code next.

P.S. If there’s some part of this story that I’m missing, please let me know. How many statistics professors could there be in Tel Aviv, anyway? Perhaps there’s some obvious explanation that’s eluding me.

Alternative spelling of Isaac Meilijson perhaps?

http://www.math.tau.ac.il/~isaco/

BTW, Christian Roberts posted some calculations regarding this occurence

http://xianblog.wordpress.com/2010/10/20/coincide…

I would guess Meilijson/Melechson's quote in the article is out of context

If you search for him in Hebrew, he's easier to locate. The problem is that his last name is being translated in some places, and transliterated in others. Here is his (English) home page:

http://www.math.tau.ac.il/~isaco/

The department has both English and Hebrew home pages, and on the Hebrew one, the professor who is called Isaac Meilijson in English is identified as Yitzhak Melechson in Hebrew.

Interesting. I looked up the guy, and it's hard to see how he could've made this mistake. Also how the renowned Freakonomics team could've missed this one.

Next up: the incredible improbable occurrence of shared birthdays in classrooms around the country!

Maybe also the words of Prof Meilijson were lost in translation. It is very likely that the reporter (the original story was published in the Israeli news website ynet.co.il) asked Prof M a few questions, and picked up the answer she liked most. The question is not presented in the original Hebrew story.

I think misquote as well. He has actually written on the birthday problem :P

I. Meilijson, Newborn, Tenenbein and Yechiali

Number of matches and matched people in the Birthday Problem. Communications in

Statistics 11(3), 361-370 (1982).

Wait, how do you know it's a mistake? This isn't the birthday problem, this is like a version of the birthday problem where you require that some two kids in the class with the same birthday are sitting next to each other in line – because he said the 6 numbers have to repeat themselves "within a month".

Anyone have the parameters for the Israeli lottery? What's the range of the 6 numbers, and how often is it played within a month?

I forgot to add my counterpoint – even if it's *not* incorrect, it still might be considered misleading, as it pretty much always is to compute the probability of events after you already know they've occurred. It's just too unfair to get to pick & choose which factors you consider in the calculations.

See also De Impossibilitate Vitae and De Impossibilitate Prognoscendi from Stanislaw Lem.

I tried to understand what Yitzhak Melechson was saying when he wrote “the incident of six numbers repeating themselves within a month is an event of once in 10,000 years.”

Since 6 numbers are drawn out of a pool of numbers from 1 to 37, the total number of combination at each lottery is

inom{37}{ 6}=2324784

Over a month, i.e. over 8 lotteries (since there are two draws per week, we can assume there 8 draws per month), the probability of no identical draws is

frac{2324784 imes 2324783 imescdots imes 2324777}{2324784^{8}}=0.999988=1-p

If we call "coincidence" the occurence of the event "over (at least) two lotteries within the same month, we obtained the same 6 numbers", each month, the probability of a coincidence is p=1.204407e-05.

The occurence of a coincidence each month as a Geometric distribution, with probability p. And it is classical, following Gumbel's definition, to consider 1/p, called the "return period", i.e. the number of months we have to wait until we observe a coincidence (i.e. a repetition in the same month), since for a geometric distribution

mathbb{E}(X)=frac{1}{p}

Here, the (expected) return period is 1/12p=6919.034 years.

From my point of view, this is “the incident of six numbers repeating themselves within a month”, and this is an event of once in 6919.034 years. On the other hand the median of a geometric distribution is

ext{med}(X)=frac{-log(2)}{log(1-p)}

which means that we have 50% chance to get such a coincidence over 4795.88 years.

So was, I am not too far away from the 10,000 years given….

Am I missing something here ?

I acutally think that you are making a mistake that surprises me, you take it for granted that this professor does not know what he is talking about when all information you have is what you have read for a newpaper (and a translation of that). If you have not experienced that journalists have misquoted you, then you probably have not talked that much to journalists.

I usually like your style, but sometimes (e.g. here and with the student a few weeks back) you are coming on a bit strong given what information you have, giving someone the benefit of the doubt is a expressions that comes to mind (but maybe not for you).

Joe:

I agree with the above commenters that the mistake is probably with the journalists, not the professor. Also see my P.S. above, which I believe qualifies as "giving someone the benefit of the doubt." As to the Freakonomics bloggers: They have a platform on the New York Times and have the responsibility to check their stories, perhaps to find a quantitative social scientist they can run their ideas by, before simply posting?

Arthur: your computation is not completely exact because you cannot take the months separately and use a geometric distribution, given the overlap of draws. I ran the my blog and I find a probability for the event of no repeated draw within 8 consecutive draws over two years to be 0.999434.

sorry, I should have said “the incident of six numbers repeating themselves within a calendar month”…

I think they just make up these academics and their affiliations. That, or they introduce lots of typos. The search [melechson site:.ir] yields zero results on both Bing and Google. There's no one named "Melechson" on the web in Israel. Also no match for [melechson statistics -lottery].