*Welcome, WSJ readers! If you enjoy this post, you might also like this look at the chances of a meteor taking down an airplane, or this one about interpreting political polling data. To recreate the calculations in this and other posts, download REvolution R.*

BBC News reported yesterday that the Bulgarian lottery is under investigation because the same six numbers were drawn two weeks in a row. The lottery organizers described it as a "freak occurrence" but nonetheless a commission has been formed to investigate this irregularity. (Interestingly, there were no winners in the first drawing, but a record 18 winners in the second drawing when the numbers repeated. I wonder if that's a common "strategy" amongst lotto players?)

The BBC did seek the advice of an unnamed mathematician, who duly pointed out the chance of this occurring was "one in four million", but "coincidences do happen". First of all, I'm not sure where he got that number from: the chance of a draw repeating last week's numbers is just 1 in the total possible draws, which I make (using R) to be more like 1 in 5 million, assuming 42 balls to draw from:

> choose(42,6)

[1] 5245786

But I'd go further and claim that given all the lotteries in the world, and the number of drawings that have occurred, it was bound to happen eventually (and be picked up for a "wacky news" column somewhere). Like I've mentioned before, an unlikely event, observed enough times, becomes expected.

Let's do some back-of-the-envelope calculations for this one. Imagine observing the Bulgarian lottery for a year. The chances of NEVER seeing a repeat of the same 6 numbers from 42 in a year's worth of drawings (52 draws) is pretty easy to calculate. We'll ignore the first week (which can't repeat anything). In the second week, the chance of seeing DIFFERENT numbers is 5245785 (1 less than the total possible draws) out of 5245786. Same for the third week, which we'll multiply by the previous probability for the second week. And so on, up to 51 weeks. The actual probability is:

> ((choose(42,6)-1)/choose(42,6))^51

[1] 0.9999903

Put another way, there's about a 1 in 100,000 chance that there WILL be a back-to-back repeat in any given year of the Bulgarian lottery.

That's still pretty unlikely, but the Bulgarian lottery isn't the only one out there. Wikipedia lists 63 national lotteries (but it's definitely an incomplete list: it lacks Australia, for example). Some of them have multiple drawings, but let's just say there are 63 major lottery drawings each week, worldwide. These lotteries have varying rules, drawing 6 from 40, 42, 45 or 49 balls: let's be conservative and assume all of them draw from 49 balls (minimizing the chances of a repeat). And let's say they've all been running for 50 years. Given those parameters, what are the chances of seeing a back-to-back repeat, somewhere in the world, in that time period?

Well, the chance of NOT seeing a world-wide repeat somewhere in the world on one week (saved in object **p**) is:

> p <- ((choose(49,6)-1)/choose(49,6))^63

> p

[1] 0.9999955

Now, let's cumulate that probability over 50 years of weekly drawings:

> p^(50*52)

[1] 0.9883548

In other words, given conservative assumptions, there's just over a 1% chance of a back-to-back drawings somewhere in the world over a 50-year period. In actuality, there are many more than 63 newsworthy lottery drawings each week, and many of those use fewer than the 49 balls I assumed (making back-to-back drawings a bit more likely). So that 1% chance is certainly an under-estimate.

Yet it's is still a fairly small chance -- it's about the same as the chance of seeing all heads from 6 coin tosses. Surprising, but not exactly remarkable. But it puts the event in Bulgaria into context -- not so quite so surprising after all.

BBC News: Bulgarian lottery repeat probed

Well, compare this to the chance that someone tampered with a lottery system in any of the world-wide lotteries in the past 50 years... I think this is more likely, although difficult to quantify exactly.

Posted by: Francois Botha | September 17, 2009 at 11:23

That's assuming that the goal of tampering would be to reproduce last week's numbers. I'd posit that that would be the *least* appropriate thing to do if you wanted to commit fraud and not raise suspicions.

But good point, and well taken.

Posted by: David Smith | September 17, 2009 at 15:26

The book "Luck, Logic and White Lies" describes how, some time ago, the German lottery had the same winning numbers as the Dutch lottery a week before. Apparently, many German lottery players pick the winning numbers of the Dutch lottery thinking they are somehow lucky. The German lottery winners never had to divide the money between so many people.

The moral of the story was that there is no winning strategy for a lottery. In fact, if you choose any one strategy together with a number of other people, your chances of winning are the same as without a strategy, but the money you'll win is (much) less if you win.

Posted by: Filip van den Bergh | September 18, 2009 at 11:39

Law of Large Numbers - given a large enough sample size any combination is possible no matter how improbable.

Words to live by.

Posted by: Brian | September 18, 2009 at 21:13

@Brian - I don't think that's really what the law of large numbers is about. http://en.wikipedia.org/wiki/Law_of_large_numbers

Posted by: Francois Botha | September 21, 2009 at 06:29

Hey! Must have been one of those "Black Swan" thingies.

Posted by: Jack | September 23, 2009 at 22:17

Isn't there a video of the numbers being drawn? Were they drawn in the same order? I assume not but I haven't heard.

Posted by: Ken Williams | September 24, 2009 at 08:46

According to BBC reports, the six numbers were drawn in different orders in

the two otherwise identical draws.

On Thu, Sep 24, 2009 at 8:46 AM, wrote:

Posted by: David Smith | September 24, 2009 at 08:48

I was trying to explain something like that to my fellow countrymen (I am Bulgarian). Provided that there are many lotteries around the globe, the chances of such a repeat are not so unlikely. It is actually a question of when and where, rather than if. And wherever it happens, people will think that there is some kind of trickery. It just happened to be in Bulgaria. If it was in South Africa, people would probably react the same way.

As a matter of fact, lottery games in Bulgaria have multiple drawings (three) per game and there are two games per week (one midweek and one weekend). The repeat happened in two consecutive games, but not in two consecutive drawings.

So, the right question would be:

"What are the chances of a repeat for an event with probability 1/5245786, if we make six trials?". I think that we should calculate this using binomial probabilities:

http://stattrek.com/Tables/Binomial.aspx

And the result we get gives us the chances for a repeat of this particular combination. However, we should have in mind that there are more than 5 million combinations and that we would have pondered on a repeat of any of them, and that we should actually estimate what are the chances of a repeat of any combination, not of this particular one. It is still small, but not so small, after all. And, having in mind that lottery games have existed for decades in Bulgaria and many other countries, the chances for a repeat go higher.

Posted by: Slex | September 26, 2009 at 01:03

All the calculations omit the fact, that the former chief of the Bulgarian lottery is one of the richest women in Bulgaria.

Posted by: Ivan Ivanov | September 26, 2009 at 05:16

@Ivan Ivanov

Rich she is, of course, but she is no longer in charge of the national lottery. Plus, if anyone wanted to make money out of the lottery, he/she wouldn't do it in such a lame way to attract so much attention.

There may be something wrong there, but those who claim that the two repeat drawings are evidence for this, simply have no case - it can happen by chance and it is not as improbable as it sounds at first.

Posted by: Slex | September 26, 2009 at 09:46

David Smith hits the nail on the head with his observation "given enough opportunity, unlikely events will happen just due to chance". However, his back-of-the-envelope calculations about the Bulgarian lotto can be refined somewhat. Let's take as starting point that in the past there were 2000 drawings in the Bulgarian 6/42 lotto (two drawings per week over two years). It is not difficult to argue that there is a probability of 0.000317 that there will be two consecutive drawings with the same six numbers somewhere in the next 2000 drawings of the Bulgarian lotto. The reasoning is based on the fact that the lotto problem can be seen as a variant of the classical birthday problem , see also the story of the coincidence in the German lotto in my book Understanding Probabilty (Cambridge University Press, second edition, 2007). In the 6/42 lotto there are 42 over 6=5,245,788 possible outcomes for a single drawing. The probability of having at least two drawings (not necessarily consecutive) with the same six numbers in the coming 2000 drawings of the 6/42 lotto is the same as the probability that in a randomly composed group of m=5,245,788 person on a planet with d=2000 equally likely possible birthdays there are at least two persons with the same birthday. It is well known that this probability can be calculated as

1-exp(0.5*m*(m-1)/d)=0.317. This probability of 0.317 can als be seen as the probability that exactly two persons among the 2000 persons share a same birthday (the probability of three or more persons sharing a same birthday is given by

1-exp((1/6)*m*(m-1)*(m-2)/d^2)=0.00005 and is thus negligible). Given that among the 2000 drawings there are two drawings with the same six numbers, there is a probability of 1/1000 that these drawings are consecutive. This gives the (exat) value of p=0.000317 for the probability that there will be two consecutive drawings with the same six numbers somewhere in the next 2000 drawings of the Bulgarian 6/42 lotto. As David Smith wrightly points out, we should take into account as well the fact that there are many lotto's in the world. Taking hundred 6/42 lotto's, there is a probability of 1-(1-p)^100=0.031 that in some lotto there will two consecutive drawings with the same six numbers somewhere in the next 2000 drawings.

Posted by: henk tijms | September 28, 2009 at 01:47

A slip of the pen in my comment. The numerical values of m and d should be interchanged, that is, you have a birthday problem on a planet with m= 2000 persons and d=5,245,788 equally likely birthdays. Otherwise, the results given are correct.

Posted by: henk tijms | September 28, 2009 at 06:33

Thanks for the refinement, Henk! Interesting approach, going about it via the Birthday Problem. I was conservative in my "world lotteries" estimate, not knowing the rules of all the lotteries worldwide. The probability doesn't increase as much as I would have guessed assuming 6/42 games as opposed to 6/49. But intuition often fails with probability, as we've seen.

Posted by: David Smith | September 28, 2009 at 09:19

People Believe in probability.It is necessary for everyone to live in support of probability .In today' world no one is perfect.this article is extremely interesting.It is valuable and helpful for one's life also.I agree comment of slex.I read this artical briefly,i have a great feeling towards it.I am also interested in this lottery and all that.

Posted by: adaptateur | October 12, 2009 at 00:12

There was more to the story, right?

In the second week following the first drawing, I think it was 3 of the same 6 numbers were selected yet again.

Also, 4 of the 6 numbers were the ones featured in the American television show, LOST.

Does this prove anything? No. But if it was a coincidence, it was even weirder than it is advertised in this column article / blog post.

Posted by: Chuck | January 15, 2010 at 15:28

I recently came across your blog and have been reading along. I thought I would leave my first comment. I don't know what to say except that I have enjoyed reading. Nice blog. I will keep visiting this blog very often.

Margaret

Posted by: Margaret | February 08, 2010 at 22:33

Can maths explain why this chance event resulted in a record number of winning tickets? Would this happen every time a set of numbers that have been drawn before are drawn again?

Posted by: Michael | March 22, 2010 at 07:34

Chance doesn't explain it, but human psychology might. I don't have any references to hand, but I've heard that there are many lotto players who play last week's numbers as a "strategy". It's similar to the documented case of a lottery having 50+ winners for one drawing. This unusual coincidence was traced to a local batch of fortune cookies (!) that all had the same set of lucky numbers printed on them.

Posted by: David Smith | March 22, 2010 at 09:11

Lol, Nobody actually questions this blogger's calculs? OMG, where did you guys learn math? All his calculs are naive and biased by his candid thinking that multiplicating will suffice and give him an accurate estimate of the probabilities of a back to back draw.. This Idiot mathematician from BBC is wrong as well, no wonder he preferred anonymity.

Unbelievable, I honestly suggest that you start to seriously question the way you do math, for your own good and that of the fools who just drink whatever you write.. Oh and a little algorithm lesson won't hurt you either

Posted by: Flatulence Daggard Washington LopezBert | May 10, 2010 at 12:07

C'mon, "about the same as the chance of seeing all heads from 6 coin tosses"?? No, it is much, much smaller chance to get the same 6 numbers in a lottery!

Posted by: Ricardo | February 15, 2011 at 12:31