## Lottery losers, raffle winners

Do you like money? Then don’t buy a lottery ticket. The total winnings will fall short of the total revenue from ticket sales (they must–otherwise, how could the government make money from the lottery?), which means buyers as a group lose money. Since there’s no way to tell if you’re the lucky buyer, the odds are that you will lose money, too.

This is the concept of expected gain. The expected gain of doing something is the average gain. It’s what you’d gain on average if you could repeat the action a million times.

With simple examples, it’s easy to calculate. You just:

1. Take each possible outcome and give it a numerical value. With the lottery, that’s really easy because the outcomes already have numerical values. The first possibility is that you win; let’s say the value is \$10,000,000. The second possibility is that you don’t win; the value is -\$1 (we’re assuming the ticket costs a dollar).
2. Multiply that numerical value buy the probability that you’ll actually get it.  Let’s say the odds of winning the lottery are 1 in 10 million, and that the odds of losing are thus 9,999,999 in 10 million. \$10,000,000 times 1/10,000,000 is \$1, and -\$1 times 9,999,999/10,000,000 is really, really close to -\$1.
3. Add together all the results. Our answer here is just about zero, though slightly on the positive side.

So the expected gain is positive! If the real lottery were like this, you should buy a ticket. (In fact, you should buy a bunch.) But obviously it’s not; the expected value to you must be negative, since the government isn’t dumb enough to set up a lottery in which it stands to lose money.

Buy the way, this doesn’t apply to raffles, or at least it doesn’t apply in a straightforward fashion. With a raffle, you win prizes, not money. Different people value the prizes differently. You might be willing to pay \$1,000 for a Caribbean cruise, while your friend might be willing to pay only \$50. Thus your expected gain from buying a raffle ticket with such a prize will differ from hers. We don’t know how much the organization running the lottery values the prizes, either; they might have gotten them at a discount that would be unavailable to you.

So a raffle might really be worth it, but the lottery isn’t.