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