How to play the lottery

by David Vose

I am a risk analyst. I am meant to know a lot about probability theory, which has its most ancient roots in the work of a colourful fellow called Gerolamo Cardano who had quite some gambling problems he wanted to resolve. If, when meeting someone socially for the first time, I say what my job is, one of the most frequent questions I get asked is whether it can help winning the lottery. Well, yes actually, it can – a bit. So – for a bit of New Year fun – I’ll share with you exactly how.

I am not the first to offer advice on how to win the lottery. YouTube videos display a wide range of methods: some are very entertaining too, inviting you to activate your dormant superhuman powers, etc. They have one thing in common – they are complete nonsense. There are also quite a number of books and software programs that you can buy that will give you fancy statistical methods for selecting the next set of balls. They are all nonsense too, I’m afraid. The UK’s Daily Telegraph1 – a very reputable newspaper – has written an article on the subject, but sadly it is tongue-in-cheek. So it appears that I may be the only genuine source of reliable information that you will get.

Lotteries differ a little, so let us take a specific example: EuroMillions is a lottery held in Europe twice a week. The price is €2 per line in the Eurozone, £2 per line in the UK, and CHF3 in Switzerland. The prizes are tax free except for in Switzerland, Spain and Portugal. The payout is about 50% of the total ticket sales.

The draw is five balls from a set numbered from 1 to 50, and another two balls (lucky star numbers) from a separate set of balls numbered from 1 to 11. The probability of selecting all seven balls correctly and therefore winning the big prize is calculated as:



Can you picture how small that probability is? If you are European, imagine a line of 116,531,800 one eurocent coins – that would stretch a distance of 1,893km, or roughly from Brussels to Lisbon. I’ll paint a dot on one of the coins and place it face down somewhere in the line – the chances of you picking that coin from the line with a single try is the chance of you winning the EuroMillions. But actually you can’t picture such a line of coins can you? Not really. You’re thinking “that’s a really small probability”, but not in a perceptibly different way than if I’d told you the line stretched in a circle around the World.

Common ‘wisdom’ suggest that we should look at how frequently the different balls have been selected during the 761-game history of the lottery. Luckily, officials have thought ahead and provide these statistics2 . We have, for draws up to 6 Jan 2015:

… which clearly and suspiciously shows strong differences in frequency – from 60 times for ball #32, to 96 times for ball #50. Yes, I have deliberately exaggerated the graph – setting the minimum to 50 – so I could inject a little gripe about how people use misleading graphs, but the 60 to 96 difference does seem rather large, doesn’t it?

In 761 games, where a ball has a 5/50 = 10% chance of being picked in each game, if EuroMillions was a fair lottery, the number of occurrences for any specific ball number should follow a Binomial(761,10%) distribution , which looks like this:

The horizontal axis in the plots shows the possible frequency for a particular ball number being selected in 761 draws, call it x. The top plot shows the probability of x occurring, the bottom plot shows the probability of falling at or below x. I’ve added 60 and 95 to the plot so you can see that there is a 2.67% chance of a frequency of 60 or less and a (1-0.9887) = 1.13% chance of a frequency of 96 or more. Those seem like quite small probabilities.

But probability math can be tricky. There are 50 different ball numbers, so 50 different and almost independent attempts at having the lowest (highest) frequency. Our ModelRisk software makes it very easy to determine the probability distributions of the minimum and maximum frequencies, using the formulae:



Running a simulation gives the following results (100,000 samples takes about 30 seconds):

… and we now see that the minimum and maximum observed frequencies of 60 and 96 respectively are precisely the most likely extremes. So it looks like the EuroMillions is fair after all.

And fair means that neither will a particular ball come up soon because it hasn’t for a while, and the frequencies need to average out to be fair – nor, the equally popular, equally wrong and opposite argument – will a particular ball come up again soon because historically it has come up more frequently than the other balls.

But, even if EuroMillions ball selection is completely random, there are some techniques that will help you do better. Take a look at this plot of the numbers that people pick:

Admittedly, these are data from a US lottery in which there are just 45 balls, but the pattern will no doubt hold. How do people select their numbers? A common approach is the date of birth of a loved one (1-31 for the day of the month, 1-12 for the month). In the chart above, 1-12 are most frequent (both day and month), 13-31 quite frequent, and 32+ less so. 7 is also a lucky number for many, which probably explains its higher frequency. There are probably other selection patterns at play too, like not selecting the start of a row, but we can see the 40+ numbers are very significantly less popular.

People also like to make a random pattern on their slip of paper. Scattering the little circles around produces a pattern that looks more realistic. However, 4,7,11,21,30 is no more or less likely a combination than 21,22,23,24,25 or 5,10,15,20,25.

If you pick a set of numbers that nobody else, or at least few others, have picked you will have to share with fewer people, so you’ll get more. In the EuroMillions the jackpot has evolved like this:

When the jackpot is won, it resets to €15 million, otherwise it rolls over to the next date. The highest number of people who have won the jackpot at the same time is five (twice), then four (three times). So if you do win, a reasonable worst case scenario is that you get just €3 million. Enough to be comfortable, but given that you actually won the lottery main prize it’s a bit disappointing and, thinking about all the people who will be coming cap in hand, the relatives, the just causes, and the cost of a decent Chateau Margaux these days, it would be better not to have to share too much.

Finally, the number of days that you have to claim varies between countries, as follows:

Don’t ask me why that is, but the EU Member States seem to like to have different rules from each other whenever possible. Anyway, this means that if you buy a ticket you have many weeks before you need to make your claim.

Summing it all up

In order to maximize your expected financial and emotional return per ticket purchased, I recommend the following:

  • Move to Austria (you have 3 years to claim, zero tax on the winnings, and pay the lowest ticket price of €2). They also have good concerts, skiing and cake there.
  • Then buy a EuroMillions ticket every 2 years and 300 days. Put it away in a very safe place. Write a note in Outlook and Google Calendar to check the ticket at the end of the 2 years and 300 days, and note where the ticket is located. This way you have the exciting feeling that you may already be a Euro-millionaire, the pride in being so full of self-control not to check, no perceivable decrease in your probability of winning than if you bought a ticket each week, and it cost you less than €1 a year.
  • Pick main numbers greater than 31, and put them in a sequence. Don’t pick 7 as a Lucky Star number – too many people think that’s lucky. Now you won’t have to share all that money when you finally do win.
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