The New(ish) Lottery Rules
I'm aware that The National Lottery has rebranded itself Lotto, but I think I'm getting old enough to be verbally conservative and refuse to accept any rebranding past my youth. (I'm only partially joking.)
In 2015 the Lottery announced that from July it was changing both its prize structure and the number of balls. It claimed you were more likely to win a prize and more likely to be a millionaire when you bought a ticket. Both of these claims were true, but since the Lottery is in the business of making money (which I suppose is also the business of business) something should ring suspicious about this. I did all of this working out last year when my friend Toby informed me of the change of rules and I include it here as an interesting probability problem and a warning against playing the Lottery, now more than ever.
OK, a quick run down of the old rules. You pick 6 numbers from 1 to 49 and then the machine (well one of them, they have delightfully named duplicates for some reason) spits out 6 balls with those numbers on. It also spits out one extra ball called the bonus ball. You win by getting at least 3 normal balls correct. The bonus ball is used if you have 5 other balls correct as a half way prize between just 5 balls and getting all 6 proper balls.
There is also a raffle that your ticket number enters you into with a shot at an additional £20,000. This is a newer addition to the structure and is the reason that the weekend players are worse off, because there are a fixed number of prizes spread over more people.
The exact amount you win is fixed for the smaller amounts and changes for the jackpot, with unclaimed prizes rolling over. However, we can say something about the expected jackpot by taking an average over time. By working out the odds of getting each prize and multiplying by the prize pool we can get the expected prize. I’ve done this twice, one for the midweek draw and one for the more played weekend draw. Here are the results:
Interestingly the payout is about 5p better midweek, but either way you are getting less than a pound’s worth of value for your ticket. This margin of profit is crazily high compared with games like blackjack and roulette.
Right, on to the changes. There are now an additional 10 balls bring the total up to 59. This sneakily lowers the odds of all the original prizes. However you can now win by matching only 2 balls which, as claimed, brings the chances of winning a prize up. Unfortunately the prize for matching 2 balls is only a voucher to buy another lottery ticket (which isn't even as good as getting your money back, because that second ticket is only worth the expected value of a ticket rather than the £1 you originally spent). However it does create some interesting mathematics, because the expected value of a ticket now partially depends on the expected value of a ticket. I'm all for recursive loops.
There are also now two levels of raffle available with the raffle jackpot giving you a fixed £1,000,000. This is why you are more likely to become a millionaire than before (just).
The easiest way to begin is to find the expected value of a ticket excluding all the 2 ball prizes. We can do that as before, so I present the midweek and weekend cases again:
However, we can match 2 balls 1/10.3 of the time, so they should influence the expected value quite a bit. We are going to have to delve into algebra at this point. The overall expected value = the expected value without the 2 balls + 1/10.3 * the overall expected value. Rearranging and doing the fraction work we get that E = (10.3/9.3) * E(without 2).
Here’s a table showing the expected value per pound that you spend for the Lottery old and new:
As you can see the odds are much worse, with the weekend pool returning you around half of your money. All this while keeping true to their word of more likely to be a millionaire and more likely to win a prize. The gambling establishment are very good at taking people’s money, don't be drawn in; but if you do, go and play something with odds only slight less than neutral. Roulette is a fool's game and even that gives you 94.7p back for every pound you spend.