In the world of business today, not much seems to be happening. Bloomberg is scrambling around trying to pick-up scoops (about Google buying out yet another map service) from “people close to the matter”. Someone wrote on article on why Economic Sanctions don’t work. There is the daily drama on what happened to US stocks and who/what the pundits are blaming.

*yawns*

So I thought I’d spice things up by explaining how easy it is to win the lottery.

And to do that, I’m going to credit one of my all-time favourite books: “Alex’s Adventures in Numberland” – which I have plagiarised before (see: Have You Really Done The Math? #SavingsAreImportant).

*Question:* How Can I Win The Lottery?

*Answer:* buy all the tickets.

*pats self on back, bows, walks off stage*

*walks back on*

*Continues*: Oh I’m sorry – did you mean “how can I make money from the lottery?”

*Answer*: **YES, YOU FOOL.**

**The Better Question: When Should I Buy All The Tickets**

Surprisingly, “buying-all-the-tickets” is not always so difficult in practice. Famously, Stefan Klincewicz, the half-Polish accountant, put together a syndicate to buyout the Irish lottery in 1992. Successfully, I might add. But the success does depend on the way that the lottery is constructed, and some time spent with permutation and combination functions on your calculator.

Most of the lotteries that I’m familiar with propose the following game of probability:

- There are 49 balls in the swirling cage
- You have to pick 6 of them.
- And it doesn’t matter what order you select them in.

This means that there are 13,983,816* possible combinations that you would have to buy in order to guarantee having the winning ticket. At the same time, I’d probably point out that there are also prizes to be won for having 5 numbers right, and 4 numbers, and 3 numbers. So if you buy every ticket, you’ll pick up those prizes as well. On average, I believe that the winning ticket gets around 52% of the “prize pool” – so we can use that to work out how much prize money is really on offer**.

Armed with that information, we can now make some fairly reasonable decisions based on the price of a lottery ticket and the size of the sweepstakes.

For example, if each lotto ticket costs £1, and the winning ticket gets £1.5 million – then that means it would cost you about £14 million to win about £3 million. Which is a complete no brainer – you go and you put your £14 million into a 1% p.a interest earning deposit account – and earn £1,150 over the three days between ticket-purchase and the lottery draw.

But let’s say that the lotto ticket costs £0.50, and the big prize is £5 million. Hmmm. So it would cost you £7 million to win about £10 million? That’s sounding more like it.

Am I missing something?

Yes. At those stakes, you’re going to get a whole lot of other people buying tickets. Probably more than another 14 million tickets… Odds are: you’ll only get half the winnings at best… That said, the size of the lesser-prizes-pool is usually linked to lottery ticket sales (I’ve seen some lotteries talk about 45% of the ticket price going into the pool) – so whatever else happens, you should get back a good part of your initial contribution. But getting back 45% of your “investment” is not a great proposition – you should only be buying up lotteries if you don’t expect to lose money.

I’m also forgetting my least favourite thing: the admin. Individual ticket sellers are going to have some limits around what they can sell, so you’re going to have to mobilise teams of ticket purchasers. And then you’re going to have to split 14 million numbers between them. When they’re done, you’re going to need to store 14 million tickets. And this is all going to have to happen in a fairly coordinated fashion – otherwise the lottery operator will pick up on what you’re doing and suddenly “*the lotto server is offline, our technicians are attending to the problem, apologies…*”

**So What Did Stefan Do?**

Stefan did indeed mobilise teams of purchasers. But he was also dealing with the Irish lottery – which only had 36 balls in the ring, taking the number of possible combinations down to 1,947,792. Each ticket cost £0.50, and the big prize was £1.7 million (owing to several rollovers from previous weeks).

So to be clear, for less than £1 million, he could buy up a lottery worth around £2.5 million***. Unfortunately for Stefan, two other people also got jackpot tickets, so he had to split the £1.7 million three ways. However, once his syndicate had also collected the smaller prizes, they’d made money. Some good money. Around £300,000 over and above the cost of all those tickets (so I’ve read).

Stefan went on to form multiple syndicates, and go in search for such prize lotteries where the size of the sweepstake was significantly larger than the cost of buying the lottery out. His general rule, in order for the bet to be a good one, was that the sweepstake had to be about three times the cost of buying all the possible combinations. From what I remember, he won a further 7 jackpots, and then retired on the money.

**The Lotteries Respond**

I can’t help but wonder whether Mr Klincewicz retired because he was bored, or if it was because the lotteries got a bit clever. Take the bonus ball story. Today, some of the big lotteries will allow the first five numbers to be in any random order, but you have to get the bonus ball (pulled from a separate drum of 36 balls) right. That is, you have to get the number right, and you have to say that it was going to be the “bonus ball”.

This takes the number of combinations up to 68,647,824. So by just playing with the rules slightly alters the odds so dramatically that you may not physically be able to buy all the tickets.

Of course this is in addition to the “let’s add more balls to the drum” and “let’s increase the ticket price” options – both of which the Irish Lottery put in place after the Klincewicz syndicate hit them.

**To Sum Up…**

You can win the lottery and still not make enough money for it to have been worth your while.

But sometimes, winning big can be your foregone conclusion.

**If you’re really interested in the math, if I just had to pick 6 balls from 49, then I would have 49 options to pick from for the first, 48 for the second, and so on – meaning there would be 49×48×47×46×45×44 different permutations of numbers. However, those permutations would include choices like 1-2-3-4-5-6 and 1-2-3-4-6-5 as two different combinations – and we know that both of those would be the same if I picked them for the lottery (ie. the order doesn’t matter). So I’d like to divide my total number of permutations by the number of different ways that 6 numbers can be combined – which turns out to be 6×5×4×3×2×1. Thus: 49×48×47×46×45×44 ÷ 6×5×4×3×2×1 = 13,983,816 possible numbers.*

***Here’s a link to the calculation of the prize pool for the UK National Lottery.*

******just the big prize was rolling over, not the smaller ones – so there was some distortion in the prize-pool*