ANYONE WHO EVER sits down to work out the probability of winning a major lottery usually tends to end up pretty disappointed at the long-shot odds of taking the main jackpot.

The odds of winning Ireland’s National Lottery are 8,145,060 to 1, for example, while the odds of winning the British equivalent are a whopping 13,983,815 to 1.

In the Euromillions, the odds get even higher – the odds of winning the biggest prize in the 12-country lottery is a scarcely comprehensible 116,531,800 to 1.

That’s to say: if everyone on the entire planet was able to enter, on average there would only be sixty winners, globally – each having to take a significantly smaller share of the jackpot.

Longtime readers of TheJournal.ie will know that every once in a while, when the biggest jackpots come along, we run over the probability of winning the biggest jackpots – and point out that it’s potentially possible to guarantee a profit whenever the Irish jackpot breaches €12.2 million.

This is because, if you had that amount of cash already, you could buy 8,145,060 tickets (costing €1.50 per line) to cover every single possible combination of numbers that would arise.

History lesson

In fact, this has happened in Ireland in the past: on the Bank Holiday weekend of May 1992, the rollover jackpot of £1.7 million was almost double the cost (£973,896) of buying every single one of the 1,947,792 combinations of numbers.

One syndicate of 28 people, preparing for the day when the jackpot would rollover, had spent six months filling out playslips – and managed to buy about 80 per cent of the possible combinations.

In the end, they shared the main jackpot prize with two other people – but the number of match-5 and match-4 prizes meant they still earned a tidy £300,000 profit. To counter this every happening again, the Lotto increased the number of balls in the draw.

While the main jackpot prize in the Euromillions has an artificial ceiling – with the 12 participating lotteries agreeing to cap the maximum prize at €190 million – a similar scheme is still possible, where buying every possible combination of ticket could bring it home.

We’ve been crunching the numbers, and extrapolating the trends from the last draw on Friday night, to try and figure out if it’s possible with the Euromillions.

So, for this example, we’re assuming you have (a) enough money to buy every single ticket possible, and (b) an elaborate system which would allow you to buy all the tickets online and not have to spend moneys filling in every possible slip.

Here are our findings.

First, a quick recap

Winning the main prize in the Euromillions requires you to match 5 numbers out of 50, and 2 ‘Lucky Star’ numbers out of 11. The odds, as we mentioned, are 116,531,800 to 1. This means buying every combination of numbers will cost you €233,063,600.

There are 13 different categories of prize – from matching all 7 numbers (5 regular numbers plus 2 Lucky Stars) down to matching 2 regular numbers and 0 stars.

If you buy every single combination of ticket, this means you’re guaranteed to win one jackpot prize (5+2), 18 of the ’5 plus 1′ prizes, and 36 of the ’5 plus 0′ prizes, and so on – all the way down to your 5,108,400 tickets which will have matched 2 regular numbers and no Lucky Stars.

All in all, of the 116 million tickets you’ve bought, 9,106,405 will be winners to some degree. But here comes the big question…

How much will they win?

Having crunched the numbers above, you’ll see there’s a 7.81 per cent chance that any individual combination of numbers will win a prize. It’s possible to take this number and use it to figure out exactly how many other people are playing.

Now, take a look at the results for last Friday’s draw – where the main jackpot of €120 million went unclaimed. Across Europe, there were a total of 4,546,468 winning tickets.

So assuming that the odds remain true (and, by and large, there’s no reason for them not to) and 4,546,468 winning tickets represents 7.81 per cent of all tickets sold, we can estimate the total number of tickets sold for last Friday’s draw at 58,179,721.

Let’s assume that Tuesday’s jackpot, given the bigger prize at stake, attracts a bumper crowd, and that there are a total of 65 million other ticket sales – you’ll still hold almost two-thirds of the total number of tickets for the draw.

Now, let’s go back again to last Friday’s draw. See the prizes for each combination? They fluctuate depending on how many other winners there are – if the jackpot goes unclaimed, the lower prizes get proportionally higher.

Fixed breakdown

For an original jackpot of €15 million, the ratios between the prizes given out at each level are fixed for each draw, with the jackpot amounting to 32 per cent of the total fund. (Full details are on page 13 of this PDF.)

If the jackpot isn’t won, its part is rolled over until next time – meaning that it becomes worth proportionally more and more of the total pot, the longer and longer it goes on.

This has a very important effect – because it means that the bigger the jackpot gets (and the closer it becomes to messing with, like this), the less and less cash becomes available to cover the holders of lesser tickets.

This Tuesday’s draw is the 15th since the last time the jackpot was won – meaning the biggest prize will account for 87.6 per cent of the total winnings.

And there, readers, is the problem. If the jackpot takes up so much of the prize fund, then only 12.4 per cent of it is left to be distributed among the winners of other prizes.

Diminishing returns

You might have 9.1 million tickets that win prizes, but each of them is going to be worth a fairly piddly amount. Assuming the main jackpot is worth €130 million (as advertised – though it’s likely to be higher) then the amount to be split between the 5+1 winners is only €1.3 million.

You’ll have 18 winning tickets, while there are likely to be an average of seven others among the 65 million tickets sold. This means the €1.3 million is split 25 ways. You’ll take 18 of those chunks, meaning €947,388 for each.

Similarly, the fund for the 5+0 winners is diminished to €433,333 – and your share would only be €302,388. Eventually, your flood of winning tickets would be so large that each of your 2+0 winning tickets, which usually win €4 or €5 each, are only worth €1 apiece.

In short, your total winnings are €140,653,079 – but you’ve already spent €233,063,600. That’s a pretty hefty loss, and it gets worse if, by fluke, someone else (improbably) wins the jackpot prize, and reduces your winnings by another €65 million.

Still, now you know – you just can’t beat the Euromillions system.