Unexpected Values, part 2

Dear Dr. Math,
Is it ever a good idea to play the lottery? My dad says he only buys a ticket when the jackpot is bigger than the odds against winning, but we're still broke.
Angelica

Dear Angelica,

NO!!

-DrM

P.S.--Here's why:

Hopefully, by now you've read my previous post all about expected values. (If not, go read it now; I'll wait here. OK? OK.) Now, the thing your dad is referring to is the fact, which is a fact, that the expected value of the lottery is occasionally greater than $1, the cost of the ticket. Assuming you play the Powerball, which is the most popular lottery in the U.S., your odds of winning the jackpot are 1 in 195,249,054. (If you'd like, I'll show you how to compute that sometime.) So, considering only the jackpot, if the payoff is higher than $195,249,054, the expected value of the lottery, i.e., the jackpot times the probability of winning, would actually be greater than the cost, so it would seem that math is telling us to play. However, this is still not a convincing argument, even putting aside other practical concerns involved in winning the lottery.*

The reason comes back to the idea of variance, which I also talked about last time. Just so you can follow along at home, the variance of a simple game like this is computed like so: you take the payoff squared times the probability of winning and subtract the expected value squared. So let's say, for example, that the jackpot was $200 million one week. Then the expected value would be the probability of winning times the jackpot, which is , about $1.02. The variance, therefore, is . Mamma mia! That's a lot of variance!

If you remember from last time, the problem with having too high a variance in a bet is that you typically run out of money before you get to win--the distribution of your winnings/losses is too spread out. So, in effect, if you had unlimited funds (which I know for a fact you don't, Angelica) and you could play the lottery with the same odds every week for several billion years, it might actually be a good investment, because on average you would win back $1.02 for every $1 you gambled, a nice healthy 2% return. However, since you're only going to play it at most a few hundred more times (and hopefully no more after today!), the variance is just too high for you to handle. It's kind of a paradox, really, that the decision to place a particular bet once depends on your ability/plans to bet many times. What we see here is an interesting example of the tradeoff between expected value and variance. Sometimes, depending on your circumstances, it's worth sacrificing a little of one to improve the other. If someone offered you $0.99 for your $1 lottery ticket, for example, I'd recommend you sell, no matter how high the jackpot is.

Incidentally, in my opinion, this is how the banking industries and insurance industries make money (or, at least, used to back when they existed). Let's say you're deciding whether to insure your $100,000 house at a cost to you of $1,500. For argument's sake, let's say you have an extra $100,000 in savings that you could use to replace the house if need be, so the only cost is monetary. And let's assume that the chance of your house being completely destroyed is ; you know it, and the insurance company knows it. So, you're trading a bet (no insurance) with an expected loss of but a fairly high variance for a sure-thing loss of $1,500 (if your house burns down, you don't lose anything except the time it takes to file an insurance claim and replace your stuff) and no variance at all. But the extra peace of mind is worth something to you, so maybe you're willing to pay the $500 premium for it. Meanwhile, the insurance company (and the bank that underwrites your policy) is buying millions of these bets, and so their distribution of income is turning into a bell curve, narrowing down pretty much to a fine point centered around a $500 profit per policy. Voilà, everyone wins, but they win more simply by virtue of already being large. (The problem occurs when they start taking that money and doubling down on risky investments, unless they get bail... oh wait, never mind.)

-DrM

*The diminishing marginal utility of money, the chance of having to split winnings with another person, the amount lost to income tax, and the failure to adjust for inflation, to name a few.