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Disclaimer: I have never bought a lottery ticket in my life.

Many people who fancy themselves mathematiclly literate make fun of people who play the lottery. Perhaps you've heard the joke, "The lottery is a tax on people who are bad at math."

Typically the argument is based on the concept of "mathematical expectation". Let me briefly explain this for those who are unfamiiliar with the concept. The "mathematical expectation" of a game or some other random event is the sum of the probabilities of each possible outcome times the value of that outcome.

For example, suppse I offered you this simple game: We roll a die. If the die comes up even, you give me $2. If it comes up odd, I give you $2. There are two possible outcomes: you win $2 or you lose $2. Assuming it's a fair die, half the time it will come up even and half the time it will come up odd. So the mathematical expectation is (1/2 x +$2) + (1/2 x -$2) = $1 - $1 = 0. What does this number mean? Well, it means that if you played the game often enough, on the average you would make $0. Your wins will tend to balance out your losses. When the total comes out to zero like this, mathematicians call it a "fair game".

Let's take a slightly more complex example. Suppose we play with this game: We roll a die. It if comes up 6, you win $4. Any other number, you lose $1. Again assuming it's a fair die, it come will up six 1/6 of the time and some other number 5/6 of the time. So the exepctation is (1/6 x +4) + (5/6 x -1) = 4/6 - 5/6 = -1/6. So on the average, you will lost 1/6 of $1, or 16 and a fraction cents, every time you play.

Note that you will not lose 16 cents every time you play. That's not possible: in each game you either make $4 or loste $1. But if you play often enough, on the average you will lose 16 cents for each time you play. If you play 600 times, you will lost about 600 * 1/6 of $1 = $100.

Again, to be clear, this does not mean that if you play 600 times you are guaranteed to lose exactly $100. You might lose $99. You might lose $102. You might be incredibly lucky, win every roll, and make $2,400. And many other possible results. But of all the thousands of possible results, the average of the results is that you will lose $100.

As the expectation in this case is negative, mathematicians would say that this is NOT a "fair game". If you play often enough, you will probably -- not certainly, but probably -- lose.

So now let's look at the real lottery. Of course the rules of lotteries vary from state to state and states often have multiple games. But just for example, here in Michigan the biggest game is "Megamillions". The chance of winning is 1 in 302,575,350. The jackpot varies, but according to the state website today it is $68 million. A ticket costs $2.

So we can do the expectation calculation. (1/302,575,350 x +68,000,000) + (302,575,349/302,575,350 x -2) = (rounding off to the nearest penny) 0.23 - 2.00 = -1.77. So on the average, every time you play you will lose $1.77.

So on the average, if you play this lottery, you will lose. The more you play, the more you are likely to lose. Therefore, it is foolish to play this game.

At which point I say, Not really. Maybe it's foolish to play, but mathematical expectation is not the definitive way to determine that.

I'm not saying that mathematical expectation isn't real or that it isn't valid. I'm just saying that, just because the expectation for some game is positive or negative does not of itself say whether that game is wise to play.

Let me illustrate with a different hypothetical game. Suppose I offerred you this game: We roll a die. If it comes up 1 to 5, I give you $25,000. If it comes up 6, you give me $100,000. You can only play the game once, win or lose. Would you play this game? Would it be wise to play this game?

By a mathematical expectation analysis, the answer is a resounding yes. The expectation is (5/6 x +25,000) + (1/6 x -100,000) = 20,833 - 16,667 = 4,166. The expected average win for you is $4,166.

But there's no way that I would play this game. Sure, it would be great to win $25,000. But I just couldn't afford to risk losing $100,000 on one roll of a die. For many people, that would mean they would lose their house, maybe everything they own. Even if you're rich enough that you could manage to come up with the cash, unless you're among the super-rich that would be a serious blow.

Mathematical expectation analysis treats the problem as purely one of net positive or net negative, without considering the probability of winning or losing or the impact of winning or losing. If I was a billionaire and $100,000 is almost what I spend on fine wines in a week, maybe I'd say, sure, I'll play this game. I'll probably win, and if I lose, oh well, too bad. But I'm not a billionaire.

Furthermore, mathematical expectation analysis treats the problem purely as a matter of arithmetic, and doesn't consider the larger context of the player's finances. If someone doesn't spend $2 on a lottery ticket, what will he spend it on instead? Will he carefully save and invest it? Or will he spend it on a candy bar, and it will be gone just as fast? Or put it toward buying cigarettes, and not only squander the $2 but give himself lung cancer on top of it?

Every now and then I hear someone say, "Oh, wow, the lottery jackpot is up to $100 million! I don't normally buy a ticket but I will now!" And I think to myself, "Really? So when it was only $10 million that wasn't worth the trouble of buying a ticket, but $100 million, now we're talking real money?"

Suppose the jackpot in the Michigan lottery was increased to $605 million, while the odds of winning remain 1 in 302 million. Then the mathematical expectation would be positive. Would that now make it wise to play the game? Because seriously, I presume that if I won $68 million tomorrow (somehow, without buying a ticket), that would radically change my life. How much more would it change it if I won $605 million? What would I buy with $605 million that I couldn't buy with $68 million? Maybe I'm not imaginative enough or greedy enough.

Does all this mean that in fact it's a good idea to play the lottery? No. Let's go back to those numbers. The chance of winning the Michigan Megamillions lottery is less than 1 in 302 million. You could play ever week for 50 years and the chance that you'll win is still less than 1 in 100,000. The chance that you'll win is minuscule. You could surely find SOMETHING better to do with that money.

© 2021 by Jay Johansen

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