The same numbers twice...

Today's news carries this story from Bulgaria. The Bulgarian national lottery numbers were the same for two weeks in a row. This is attracting gasps of amazement from commentators and an investigation from the Bulgarian Lottery regulator. But the fact that this happened is actually not that surprising. The following explains why.

Firstly, the paper is referring to these "identical" draws. They weren't. The numbers came out in a different order each time. For those who remember their high-school maths, this is the difference between a "permutation" and a "combination". The odds of 6 balls coming out in exactly the same order twice are very long, but with lotteries, you need the right numbers, you don't care in what order the balls emerge. This means we use the maths of combinations.

The odds of winning first prize in the lottery are easily calculated. Lets say you are picking p balls from n possible numbers. The chance that the first ball out is one of yours is p/n. There is now one less ball in the barrel, and one less number that you need, so the chance of the second ball being one of yours is p-1/n-1. You keep going like this until you've pulled out all your balls. You then multiply these probabilities together, as all the events must happen for you to win.

For a 6 in 42 game, the odds are about 1 in 5.2 million. These are pretty ordinary odds, but as they say, somebody has to win it. (Well, actually, they don't. If nobody picks the right numbers, the first prize will never go off.)

But what about consecutive draws? The chance of picking the right numbers is exactly the same as the chance of the same numbers popping out two weeks in a row. All we are doing is picking 6 numbers. It just so happens that, in this case, the numbers we are using are the numbers that came out of last weeks draw. Despite all the books published detailing the least and most often drawn balls, this data is irrelevant for a given draw. Every ball is equal. To use our example, the odds of two consecutive 6 from 42 draws yielding the same results is 1 in 5.2 Million.

But really, the odds we are looking for are not the odds of winning this lottery. We are looking for the odds of creating this news story. These are the odds that, somewhere in the world, at some time when there is news, two consecutive lottery draws yield the same results.

Firstly, we don't care when this happens. This news story could have been last week, or next week. All we need is two consecutive draws, so any weekly game will do. There are 52 weeks in a year, each of which has a 1 in 5.2 Million change of repeating all numbers. As we only need one of these events to happen, we add the probabilities to give 52 in 5.2Million, or 1 in 100,000.

Secondly, We don't care where this happens. Honestly, Bulgaria? If it wasn't for the Monty Python song and this lottery coincidence, it wouldn't be in the news. Had it happened in Estonia, Lithuania or South Africa, the news would have been reported.

A quick Google suggests that there are around 250 lottery commissions around the world. Each US state has one. Each Australian state has one. Most European countries have at least one. Most commissions offer multiple games, some with many draws per week. For example, NSW state lotteries has OZ lotto, Powerball and Lotto, with the latter drawn on Mondays, Wednesdays and Saturdays.

There is a research project in doing this properly, so I'm going to switch to hand waving mode.

Its reasonable to guess that between the 250 lotteries commissions we know about - and those that we may not - we have 1000 weekly draws. We know the chance of 1 lottery doubling up is 1 in 100,000. We now have 1000 chances. 1000 in 100,000 is 1 in 100.

1 chance in 100 that somewhere in the world, in a 12 month period, a national or state lottery will draw the same numbers, two weeks in a row.

Those are remarkably good odds. We can speculate that the odds are actually slightly better than this. Coincident draws of the same numbers in nearby countries or states would also generate a news story. As would draws with different time separations (like NSW Lotto on Monday and Wednesday). As the human brain is geared towards looking for patterns, when they occur, they are almost certain to turn up in the news.