Practical Probability - Is the lottery a 'tax on the stupid'?

The following is a post that deals with some fairly basic probability theory, and addresses the question of whether it's fair to say the lottery is a 'tax on the stupid'. It's mostly aimed at anyone who hasn't done maths past GCSE, since the ideas won't be particularly surprising or interesting to anyone whose thought about probability beyond what they were forced to do at school.

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I have often heard people refer to lotteries as a 'tax on the stupid', and on first glance it's hard to disagree. Even without knowing about the probabilities involved, like any gambling method the house always wins, and so as a player you're expected to lose.

We take as our motivating example the UK National Lottery, rebranded as 'Lotto' a few years ago. If you buy a Lotto ticket, what can you 'expect' to win? Before we calculate this, we take a very brief detour into some basic probability theory, to illustrate how to work out expected winnings. This will probably be familiar to most people reading this, so some skim reading might be in order for some of you.

First, let's just talk about the probability of something happening. Suppose I toss a coin and ask you to call heads or tails, you might say that your chances of getting it right are 50/50, 1 in 2, or 50%. I'd say your probability of getting it right is 0.5, and it's this way of saying it that we'll stick with.

When talking about probability in this sense, we use a scale from 0 to 1, where 0 means there's no chance of something happening (like the probability of rolling a die labelled 1 to 6 and getting a 7) and 1 means that the outcome is definitely going to happen. A probability of 0.5 is halfway between 0 and 1, and so indicates an outcome that is as likely to happen as it is to not happen.

Now, suppose we play a a simple gambling game. I toss a coin and you stake £1 on the outcome being heads or tails. So if you are betting £1 on this outcome, what is a fair return if you win? You would probably say instinctively that it's fair if you profit £1 if you win, since you lose £1 if you lose. In other words, if you win I should pay you £2 (including the £1 you gave me to start with), and if you lose I give you nothing. We can check this instinctive guess is in fact right by doing some pretty simple algebra.

Let's say I give you £x if you win, and otherwise I keep your £1. There are two possible outcomes to the coin toss:

You win the toss with probability 0.5 and profit £x - £1
You lose the toss with probability 0.5 and lose £1 (in other words, you 'profit' -£1)

You can work out your 'expected' winnings by multiplying the probability of an event by what you profit, and then adding these up for all the possible outcomes. In this case there are just two outcomes, and so the expected return is 0.5*(x-1) + 0.5*(-1), corresponding to your 0.5 probability of profiting x-1 pounds, and the 0.5 probability of you losing 1 pound. If we expand the algebra we get 0.5*x - 0.5 - 0.5 = 0.5*x - 1.

A bet is 'fair' if your expected profit is zero, i.e. if in the long run you would expect to neither win nor lose money. So to choose x to make the bet fair we have to find an x so that 0.5*x - 1 = 0, and it's not too difficult to see that x = 2 satisfies this condition. As we guessed, the game is fair if I give you £2 back if you win.

So anyway, back to the lottery. To calculate one's expected return from a lottery ticket we can just apply the above probability theory to the slightly more complicated lottery prize structure, right? Well, not really. Whilst it's easy(ish) to calculate the probability of each winning combination (matching 3, 4, 5, 5 and the bonus ball, or all 6 numbers), the prizes you get for each outcome are variable. The wikipedia page about the lottery details the precise mechanism, as well as telling you the probabilities of each outcome, but the important point is that only the £10 prize for matching three numbers is fixed. The other amounts are determined by how much money is left in the prize fund once all the £10 winners are accounted for. (This is why when you see a draw on TV they talk about the 'estimated' jackpot; they don't know what the final jackpot will be until they know how many £10s have been won.) Another problem is that how much a ticket wins depends on how many other people win that prize, which complicates the expectations even more.

Fortunately, there is an easy way to work out the expected return for a ticket. From all the money made from ticket sales, Camelot set aside 45% for the prize fund, with the rest going to charities and tax (as well as a profit for the company). Every ticket has the same chance of winning as every other ticket, and so the expected return for every ticket must be the same. Because 45p from every ticket is then given back to the people buying tickets, this means that your expected return from a £1 ticket is that 45p. In other words, for every £1 ticket you buy, you can expect to lose 55p, in the long run at least. That's a pretty terrible return, so maybe that 'tax on the stupid' line isn't too inaccurate after all.

In fact, as gambling games go, Lotto is one of the more 'unfair', at least in terms of the punters' expected returns. For example, on an American roulette wheel, there are 18 red, 18 black and 2 green numbers (the 0 and 00). If the green numbers weren't there, then betting on red would be like betting on a coin toss, and so a fair payout on a bet of £1 would be £1 as we calculated earlier. Of course, in real roulette the payout isn't fair, and whilst you do get double your stake back if you bet on red and win, the two green numbers make winning slightly less likely than it should be for this to be a fair bet. More precisely, if you put £1 on red, then your expected return is (roughly) £0.95; on average you 'only' lose five pence per spin. Compared to the £0.45 you get from a Lotto ticket, the roulette wheel seems like a great deal.

So anyway, if you gamble, you're expected (on average) to lose, so it's stupid to do it - is that a fair assessment? You can probably guess that I'm not convinced it is. Calculating 'expected' returns based purely on probabilities makes one fairly major assumption: the value of money is linear. That is, it assumes that the difference between £50 and £100 is the same as the difference between £1,000 and £1,050. You're probably thinking "well it is, it's £50 both times", but that's not quite the point. For instance, suppose someone calculated the most money you could ever possibly want or need in your lifetime, and then someone else offered you double this. Calculations about long-term expected returns assumes that the second offer is worth twice as much as the first, and whilst that is obviously the case in terms of raw numbers, to an individual there isn't really any difference. If I've got as much money as I could ever possibly want, then any more money is worthless to me.

If this example is a little too fanciful for your tastes, then let's construct a moderately more realistic scenario. If someone offered you £50 or £100, you would take the £100 without question, and feel much better off for it. On the other hand, if someone offered you £1,000,000 or £1,000,050, you'd probably still take the larger amount, but that extra £50 seems far less valuable, because compared to £1,000,000 it's virtually nothing.

The point here is that the value of money is not simply how many zeroes there are on the end of a number, and so calculations of expected lottery returns are a bit meaningless if you don't take this into account.

So if we can't calculate value of a lottery ticket by just multiplying all of the possible winning amounts and the probability of attaining them, what can we do? Well we can still use this method, it's just that we have to be a bit more careful in how we define what the winnings are. Instead of calculating one's expected return in terms of pounds and pence, we have to instead think about the 'value' of each possible outcome. Investigating what money is worth to people is an area of psychology/economics which has received a fair bit of attention, and as you'd expect it's the sort of thing which can vary enormously from person to person. It's worth bearing this in mind, then, the next time you hear someone call lottery players (or even gamblers in general) 'stupid'; they might have thought about it a bit more than you think.