Straightforward statistical arguments would suggest that you should not play a lottery as the odds are weighted such that on average you always lose. However, a lottery is a good example where common statistical arguments are in fact useless. I will argue that it is completely rational to play a lottery –but only if you want to– and will conclude with a remark on climate change.
First of all, how does a lottery work? A lottery essentially gathers small sums of money from a large number of players and then hands back a large sum of money to a small number of players. Of course, the amount handed back is less than the amount gathered, so the lottery organiser earns money doing so. National lotteries use this trick to extract some extra cash from its citizens and usually hand out some amount to charities.
The National Lottery in the UK earns 12% in lottery duty for the UK Government, 57% is handed out as prizes, 24% goes to charities and good causes, 3% is commission for the shops that sell lottery tickets, and about 4% is used for operating the whole shebang. The annual ticket sales in the UK are about 8 billion pounds (data from year ending Mar 2022.)
Let us simplify this system to a lottery with 5 people, each buying a 1 pound lottery ticket and the winner gets 3 pounds. The organiser gets 2 pounds. The percentage of ticket sales going out as prize money is similar to the UK National Lottery.
It does not seem rational to participate in this five-person lottery. Of the money put in by the participants, 40% is lost every round. There will be one lucky winner making a two pounds profit but there are four times as many losers. So if you play this lottery regularly you are bound to lose on average 40% of the money you put in.
Statisticians call this an expectation value: there is a 20% chance of winning 3 pounds, 80% chance of not winning anything, so the expectation value of the earnings after one round are 0.2×3 + 0.8×0 = 0.6 pounds. Each round costs 1 pound, so on average you expect to lose 40 pence every time you play.
This just seems dumb. Why would anyone want to play this lottery?
But what happens if we have 10 million players rather than five players. Let us use the same winning percentage: of the ticket sales, 60% gets distributed to the winner. The calculation of the expected earnings for each player leads to exactly the same outcome: it is 60 pence of prize money for every pound you put in, so you expect to lose on average 40 pence every time you play.
[In the calculation, the 20% chance of winning in our five-people lottery reduces by a factor of 2 million, but the earnings for the winner have increased by a factor of 2 million as well. In the UK National Lottery, the winning prize money is less, but your chance of winning it is higher –there are many prizes handed out each draw.]
So common statistical argument tells you again that you are bound to lose 40% of your money on average when you participate in this lottery, so it seems equally irrational to participate in this big lottery as it is for the five-person lottery.
This is where statistics based on expectation values fall apart.
Just imagine you would play this 10 million-people lottery once a week. Most people would not be concerned about randomly losing one pound each week. The cost of participating in a five-people lottery or a 10 million-people lottery is the same. The expected earnings are also the same: on average you are going to lose 40% of that money, although the vast majority of people will lose 100% of their money. However, in the unlikely event of winning, you win a life-changing amount of money. This is different from the five-people lottery: winning three pounds is as irrelevant for most people as losing one pound.
So the vast majority of those 10 million lottery players will simply lose their one pound every week. But for none of them this is a measurable loss. (I realise that sadly this is not true anymore for quite a few people these days.) So effectively the participation in our weekly lottery does not have any measurable cost to the participants, but it does give you a tiny chance to earn a life-changing amount of money, and a weekly thrill when the lottery winner is announced.
Playing a lottery is very different from going to a casino. In a casino the expectation value for the earnings is in fact much higher than for a lottery. Depending on the “game” played, a casino will return more than 90% or 95% of wagers as winning payout. So the statistical argument for playing casino “games” seems substantially stronger than for playing a lottery.
But (there’s a massive “but” here) participation in casino “games” costs a non-negligible amount of money. So even though you will get a better payout percentage, you are essentially losing whatever the expectation value tells you. Perhaps you are “only” losing 5% of your wagers every time you “play,” for most people that simply means that you continue to “play” until you have lost the money you were willing to spend in the casino. You just keep re-cycling your earnings until they have eroded completely, by 5% for each “game” on average.
How to explain this statistically? Expectation values rely on sampling all possibilities. In a casino you would typically play many “games” (often by spending your winnings as wagers for following games) and therefore you would sample all possibilities for winning and losing. But the casino has the edge on the player: on average the casino is to gain 5% or so of all wagers.
In a lottery the chance of winning is very small, so any single player will not sample all possibilities for winning and losing, and the expectation value for earnings is an irrelevant statistic. You are very likely to lose a small amount of money or you may win a very large sum of money. The expectation value reflects the average outcome of these two very distinct possibilities.
This is an example of where an average actually does not describe a realisable situation. Think about the average location of the ball on a roulette wheel: the average position of the ball is exactly in the centre of the wheel, something that is never realised.
Enough about lotteries and casinos. What about climate change?
Climate is a good example of an unrealised average. The climate is a statistical construct, an expectation value of meteorological parameters. But the climate itself does never actually happen. In a similar vein, global mean temperature has very little meaning, and I have written before about other ways in which global mean temperature has no physical meaning.
Understanding future climate is an even trickier concept. We have all kinds of future scenarios to consider when we try and understand what a future climate might do. Much of the statistical analysis of these climate scenarios is based on expectation values across these scenarios (so-called ensembles in climate modeling). But, of course, only one scenario will ever be realised. Should we base prevention and mitigation strategies on expectation values of climate scenarios or only on the most likely scenario? In fact many people argue we should take a cautionary approach, by mainly considering the more extreme scenarios.
There seems to be no easy statistical answer to such questions, although arguments based on expectation values seem wrongheaded.
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