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What are the odds?

Opinion and Analysis

Is it really that unusual for a well-run lottery to produce consecutive sets of identical numbers?

To answer this question, we should try to set some context.

As far as we know, this has never happened before.  Surely, it would have been all over the news (as this story is) if it had happened in the past. 

The only similar story I could find was for a US-run lottery where three digits are randomly selected, giving a number from 000 to 999 - once, this produced the same 3-digit value three times in a row.  The odds for this 3-number run are just 1:1 million.  Considering that such lotteries typically cycle every few minutes, this is quite unsurprising.

For 6-number lotteries, of course the odds are much higher.  However, consider this (based on numbers 'out of my head' but seeming to be reasonable): there are probably at least 500 such lotteries running around the world, generally with at least 2 draws per week.  Pretty-well all of these lotteries have been running for 20 - 40 years (lets use an average of 30 years).

Bringing these numbers together, the world is likely to have experienced somewhere around 1.5 million such draws.

Think about it.  1.5 million attempts to hit odds of 5.24 million to 1 suddenly seems quite 'reasonable.'

There's one final oddity in the Bulgarian incident.  The original draw had no winners, yet there were 18 winners the second time around.  Observers pointed to this fact to suggest that there was some kind of scam going on.  However, according to most reports, each of the 18 would expect to receive a prize approximately equal to the typical yearly wage in the country.  Hardly reason to set up a complex, very public scam such as this.

Also, psychologists will tell you that regular lottery players will tend to shy away from the numbers in a previous draw on the basis that "surely they won't be drawn again."

Perhaps this means that Bulgarians are a lot more logical (or fatalistic) than most people.

Either way, this is an interesting exercise in statistics and how people's intuition can lead them astray.

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