Lotteries take place in many places around the world, and are an interesting case study in human behaviour and our relationship with mathematics. It is fascinating how something which in most cases has an incredibly small chance of happening is such an attraction for players – the probability of winning the lottery jackpot is conventionally one in tens of millions. People continue to buy tickets, drawn by the tiny chance of a life-changing amount of money.

Since lotteries are usually drawn using a completely random method in the interests of fairness, it is definitely not possible to increase your chances of winning by choosing particular numbers. However, it is possible to increase the size of your potential prize through your choice of numbers.

In lotteries where the top prize is split between everyone who matches all the numbers, you are playing against everyone else, as well as the random nature of the draw. In these situations, knowing which numbers are more likely to be popular is useful – generally, this tends to include numbers locally considered “lucky” – as is avoiding unlucky numbers like 13 (in Western cultures), 4 (China, Japan and Korea), 9 and 43 (Japan), 17 (Italy) and 39 (Afghanistan).

A survey by Alex Bellos back in 2014 found that 7 is considered to be a lot of people’s favourite number, so that is likely to be popular; and many people choose their numbers based on their own and family birthdays, so numbers between 1 and 31 tend to be more frequently chosen than numbers above 31. So potentially, picking numbers some consider unlucky or are otherwise less likely to choose means that if you do win big, you are less likely to share the jackpot with as many others.

Gaming the System

Of course, all of this is subject to the same annoying probabilities – your chances of winning are still vanishingly small. Is there anything we can do to make it more likely – or even guarantee – that we will be a lottery winner?

Theoretically, it should be possible to buy as many lottery tickets as you want – with a sufficiently coordinated network of buyers and probably quite a big spreadsheet, every possible combination of numbers could be calculated, allocated and purchased. The cost of doing so would be – with exact values subject to the specifics of the lottery you are playing – a significant investment, but has been theorised as a way of guaranteeing yourself a win. You would definitely have a ticket that matches all the numbers!

But the numbers do not necessarily stack up if you calculate the expected return on this technique. For example, the UK lottery, when it started in 1994, used six numbers drawn from 49 and had tickets costing £1. From this we can work out the expected return of this strategy, by looking at how much it would cost and how much you would win.

The number of possible sets of numbers you could choose can be calculated as 49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520, since each successive number is drawn from a slightly smaller set of options. We then need to divide by 6 × 5 × 4 × 3 × 2 × 1 = 720, as we do not care what order the balls are drawn in, and this is the number of ways to rearrange them (my previous post on combinatorics covers this in more detail).

This gives us 13,983,816 possible combinations (and also tells us that the probability of winning the jackpot by matching all six numbers is 1 in 13,983,816, or 0.000007% – less likely than being struck by an asteroid). The cost of buying one ticket for each would then be just under £14 million; but how much would you be likely to win?

Things get slightly complicated here, since the UK lottery, like many others, has prizes for matching fewer than six numbers, and within your collection of tickets you would also have all of these possible outcomes too (although you may wish to outsource taking the 246,820 tickets that each match three numbers down to the shops to claim £10 for each of them). You would also win prizes for matching 4 and 5 numbers; and the UK lottery also has a “bonus ball” drawn separately as a seventh number, which comes into play when you have also matched 5 of the main numbers, and gives you an extra prize.

The size of the jackpot itself depends on the number of tickets sold – and admittedly, a shadowy group of mathematicians purchasing a huge chunk of tickets for one particular draw would increase this amount fairly significantly. However, if you run the numbers on an average weekly draw under the original prize system, from the jackpot and all the smaller prizes together you would net a total of around £6,292,717 – less than half what you spent on tickets (not to mention logistics). Of the 14 million tickets you bought, only 260,624 would win any kind of prize – of which the bulk would be your £10 tickets – and the rest would go straight in the recycling.

The UK lottery also operates a “rollover” on occasions when nobody wins the jackpot – a portion of the prize fund that was set aside for the jackpot gets added to the jackpot for the following week. But even when there is a rollover, a double rollover (when this happens twice in a row) or a triple rollover, you are still expected to lose a good chunk of your £14m, with the amount you can expect to win being still less than £10m.

More recently, this has gotten even worse: the UK lottery has increased its number set to picking 1 out of 59 – now meaning there are 45,057,474 combinations – and tickets now cost £2 per draw, so you would have to put in nearly £100m to play all the tickets, and would definitely make an even bigger loss – the average maximum jackpot is only £13m (with a cap on how many times it can roll over), and smaller prizes would not add much to that total.

It makes sense that it is not possible to make a profit doing this – on the UK lottery, for every £1 ticket purchased approximately 45p went into the prize fund, whilst the other 55p goes into charitable donations – as well as covering operational costs and creating a profit for the company running the lottery. So there is no way you could ever get out more than you put in – like any gambling-based activity, if someone is making money out of running it, you can always expect to lose out overall.

Mathematising the Game

This does not mean there is no hope at all, though: people do still win the lottery, and net huge amounts of money. With any gambling game, it is always a possibility that you will walk away with more money than you spent – if you are lucky. While you cannot control fortune, it is possible to change the way these probabilities interact, mathematically, by making careful choices.

For example, when playing roulette, your options include betting on a particular colour (around half the slots on the wheel) or betting on a number (one of the slots); you win a lot more money for correctly matching a single number, but you can control through this choice based on whether you would prefer to have a higher chance of winning a bit of money, or a lower chance to win a lot – you can think of this as ‘concentrating’ the probabilities in a way that suits you.

The good news is, if all you care about is the buzz of winning, some mathematicians have now found a way to guarantee this for you. Some researchers at Manchester University recently released a paper entitled “You need 27 tickets to guarantee a win on the UK National Lottery”, which uses clever combinatorial arguments to suggest a small set of tickets which between them will definitely match at least two numbers, winning you a small prize under the current system (a free random play on the following lottery).

The system involves using a clever piece of mathematics called finite geometry – I have discussed it here previously when talking about block designs. These are structures that include all the possible combinations of objects from a given set of sets – in my previous post, this was about arranging objects with given properties into groups. The diagram below shows these structures represented as diagrams – in the bottom three, which are a structure called the Fano Plane, each line contains three points, and each point lies on three of the lines (one of which is not a straight line, but we will forgive it). 

Having assigned two numbers to each of the points, buying a ticket with the six numbers attached to each line, and creating planes that cover all the numbers, guarantees that for the six numbers in the lottery draw, one of your tickets definitely will contain at least a pair. The other two diagrams are included to make sure all 59 of the UK lottery’s numbers are covered – 3 lots of 14 numbers on Fano planes makes 42, and we could just use a fourth to include the rest. In this case, the remaining numbers are instead arranged in simpler structures to let you buy the minimal number of tickets – proven in the paper to be 27.

You can of course choose to assign the numbers differently to the way it has been done in the diagram – and it is probably a good idea to do this, since the paper has been published and I imagine there are a non-trivial number of people using the numbers suggested in the paper, so – as we have seen – avoiding those specific ones will decrease your chances of sharing a prize. (For a more in-depth explanation, Matt Parker has made a video, in which he goes and buys the 27 tickets for one week’s draw – you will have to watch to see what happens.)

The problem here is again that you should not expect massive returns: your “win” might just be a free extra play (which is notionally worth another £2, but as we have discussed, is unlikely to be worth much more than that) – and even matching three numbers, which is not guaranteed here, only nets you £30. Since this is an example of a way to manipulate and combine probabilities, guaranteeing yourself a minimum of a two-number match will slightly reduce your chances of making a three-number match if you play these tickets instead of 27 random draws (around 23.5% instead of the usual 24.4%).

Peter Rowlett at The Aperiodical did some quick simulations, creating all 45 million possible lottery draws and counting which ones would win you enough to cover your initial investment (£54, for 27 tickets at £2 each). The probability of making a profit, based on the proportion of all possible draws where this happened, is less than 5% – and in around 75% of draws the only win was a match of two numbers. So maybe this is not a guaranteed path to fortune – but it is an interesting bit of maths!