Now that we’ve discussed Pascal’s calculations of “expected value” and the various shenanigans the Lottery uses to make the prize look bigger than it really is, let’s talk about the end results of being a so-called winner.
First, ask yourself what money is good for. Seriously. There are things you can get if you have money which you can’t get if you don’t have money. For example, if you have $2 you can get a hamburger, which can mean a lot if you’re very hungry. If you have just $10 you can get a good meal at a restaurant. If you have $50 you can spend the night in a hotel room instead of sleeping on the street. If you have $1,000 you can rent an apartment for an entire month. With $10,000 you could buy a nice used car or an older motorhome. With $100,000 you could buy a small house or a really nice motorhome. With $1,000,000 you could buy a really nice house plus have enough money left over to buy food for yourself for several years. Each of these examples demonstrates the utility of money. With it, you can get something you need or want, without it you can’t. This might affect your happiness level (or might not) but it certainly can affect your health and your safety.
But how much utility can you get from $2,000,000 compared to $1,000,000? You can buy a house that’s twice as big. But will that make you twice as happy? Will it make you twice as safe? Will it make you twice as healthy? Certainly not. The difference between struggling to find food and shelter vs. having a nice house full of food is about $4,000 per month. Beyond that, having more and more money only adds a tiny amount to the list of things you can do and how healthy/safe/happy you’ll be.
Of course, if you got $10,000,000 you could give most of it away to other people. Then both you and 9 of your friends could each have a house full of food. But my point remains that $10 million in the hands of one person is not 10x better than $1 million.
Consider the following game. I’ll put 30 six-sided dice into a shoebox and shake it up. You buy a ticket from me for $1,000 and then we open the box. If all 30 of the dice have landed on 6, I pay you $1,000,000,000,000,000,000,000,000,000. That’s one billion billion billion dollars, also known as an octillion. It’s many many times all the money on Earth right now. If you had that much money, it would literally be impossible for you to spend it all because there simply isn’t enough stuff on Earth for you to buy. Let’s calculate the expected value for this game. The chance of rolling 30 sixes is 4.52337×10^-24, which is .00000000000000000000000452337 .
EV = (4.52337×10^-24)x($1 octillion) – (1-4.52337×10^-24)x($1,000) = +$3,523.37
As you may remember, any positive expected value at all means that the game is tilted in favor of the player and it’s a “good” bet. In this case, you are risking $1,000 and expect to make a profit of $3,523.37 which is a fabulous return on your investment. Would you do it? Would you actually pay me $1,000 for a ticket to play this game?
I submit that, even if you believed that I’d be able to follow through on the promise of paying out if you win the bet, it still would be foolish for you to spend $1,000 on a ticket. The amount of happiness you’d get from winning simply isn’t worth what you give up by having to pay me $1,000.
This demonstrates a fundamental flaw in the Expected Value formula. It assumes that getting 100x as much money has 100x as much utility or will make you 100x as happy, and that’s simply not true. The larger the numbers involved, the less useful the formula becomes.
However, there’s a positive result from buying a lottery ticket which has nothing to do with winning. Just buying a ticket gives you a chance to dream about changing your life. If the amount of happiness you get from dreaming about becoming a millionaire makes you happier than keeping the cost of that ticket, then it might be money well spent.