Expected Value and the Lottery

Don’t play Powerball

The nationwide lottery game Powerball was playable, at the time of this account, in forty-two U.S. states, the District of Columbia, and the U.S. Virgin Islands. It was extremely popular, sometimes selling as many as 100 million tickets for a single drawing. Poor people play Powerball and people who are already rich play Powerball. My father, a former president of the American Statistical Association, plays Powerball, and since he usually gets me a ticket, I guess I’ve played, too.

Is this wise?

On December 6, 2013, as I write this, the jackpot stands at a handsome $100 million. And the jackpot isn’t the only way to win. Like many lotteries, Powerball features many levels of prizes; the smaller, more frequent prizes help keep people feeling the game’s worth playing.

With expected value, we can check those feelings against the mathematical facts. Here’s how you compute the expected value of a $2 ticket. When you buy that ticket, you’re buying a:

1/175,000,000 chance of a $100 million jackpot

1/5,000,000 chance of a $1 million prize

1/650,000 chance of a $10,000 prize

1/19,000 chance of a $100 prize

1/12,000 chance of a different $100 prize

1/700 chance of a $7 prize

1/360 chance of a different $7 prize

1/110 chance of a $4 prize

1/55 chance of a different $4 prize.

(You can get all these details from Powerball’s website, which also offers a surprisingly spunky Frequently Asked Questions page, filled with material like “Q: Do powerball tickets expire? A: Yes. The Universe is decaying and nothing lasts forever.”)

So the expected amount you’ll win is

100 million / 175 million + 1 million / 5 million + 10,000 / 650,000 + 100 / 19,000 + 100 / 12,000 + 7 / 700 + 7 / 360 + 4 / 110 + 4 / 55

which comes to just under 94 cents. In other words: according to expected value, the ticket isn’t worth your two bucks.

That’s not the end of the story, because not all lottery tickets are the same. When the jackpot is $100 million, as it is in this example, the expected value of a ticket is scandalously low. But each time the jackpot goes unclaimed, more money enters the prize pool. And the bigger the jackpot gets, the more people buy tickets, and the more people buy tickets, the more likely it is that one of those tickets is going to make somebody a multimillionaire. In August 2012, Donald Lawson, a Michigan railroad worker, took home a $337 million jackpot.

When the top prize is that big, the expected value of a ticket gets bigger too. Same computation as above, but substituting in the $337 million jackpot:

337 million / 175 million + 1 million / 5 million + 10,000 / 650,000 + 100 / 19,000 + 100 / 12,000 + 7 / 700 + 7 / 360 + 4 / 110 + 4 / 55

which is $2.29. Suddenly, playing the lottery doesn’t seem like such a bad bet after all. How big does the jackpot have to be before the expected value of a ticket exceeds the $2 it costs? Now you can finally go back to your eighth-grade math teacher and tell her you figured out what algebra is for. If we call the value of the jackpot J, the expected value of a ticket is

J / 175 million + 1 million / 5 million + 10,000 / 650,000 + 100 / 19,000 + 100 / 12,000 + 7 / 700 + 7 / 360 + 4 / 110 + 4 / 55

or, to make it a little simpler,

J / 175 million + 36.7 cents.

Now here comes the algebra. For the expected value to be more than the two dollars you spent, you need J / 175 million to be bigger than $1.63 or so. Multiplying both sides by 175 million, you find that the threshold value of the jackpot is a little over $285 million. That’s not a once-in-a-lifetime occurrence; the pot got that big three times in 2012. So it sounds like the lottery might be a good idea after all—if you’re careful to play only when the jackpot gets high enough.

But that’s not the end of the story either. You are not the only person in America who knows algebra. And even people who don’t know algebra instinctively understand that a lottery ticket is more enticing when the jackpot is $300 million than when it’s $80 million—as usual, the mathematical approach is a formalized version of our natural mental reckonings, an extension of common sense by other means. A typical $80 million drawing might sell about 13 million tickets. But when Donald Lawson won $337 million, he was up against some 75 million other players.*

The more people who play, the more people win prizes. But there’s only one jackpot. And if two people hit all six numbers, they have to share the big money.

How likely is it that you’ll win the jackpot and not have to share it? Two things have to happen. First, you have to hit all six numbers; your chance of doing so is 1 in 175 million. But it is not enough to win—everyone else must lose.

The chance of any particular player missing out on the jackpot is pretty good—just about 174,999,999 in 175 million. But when 75 million other players are in the game, there starts to be a substantial chance one of those folks will hit the jackpot.

How substantial? We use a fact we’ve already encountered several times; that if we know the probability that thing one happens, and we know the probability that thing two happens, and if the two things are independent—the occurrence of one has no effect on the likelihood of the other—then the probability of thing one and thing two happening is the product of the two probabilities.

Too abstract? Let’s do it with the lottery.

There’s a 174,999,999 / 175,000,000 chance that I lose, and a 174,999,999 / 175,000,000 chance that my dad loses. So the probability that we both lose is

174,999,999 / 175,000,000 × 174,999,999 / 175,000,000

Each of us individually has roughly a 99.9999994% chance of losing; the product above gives a slightly smaller probability that we both lose. In other words, as I tell my dad every single time, we’d better not quit our jobs.

But what’s the chance that all 75 million of your competitors lose? All I have to do is multiply 174,999,999 / 175,000,000 by itself 75 million times. That sounds like an incredibly brutal detention assignment. But you can make the problem a lot simpler by phrasing it as an exponential, which your computer can calculate for you instantaneously:

(174,999,999 / 175,000,000)^75 million = 0.651 . . .

So there’s a 65% chance that none of your fellow players will win, which means there’s a 35% chance at least one of them will. If just one other ticket wins, your share of the $337 million prize drops to a puny $168 million. Treating every shared jackpot as a two-way split puts an approximate upper bound on the average prize conditional on your ticket winning:

65% × $337 million + 35% × $168 million = $278 million

which is just below the threshold value of $285 million that makes the jackpot worth it. And that doesn’t even take into account the possibility that more than two people will hit the jackpot, divvying up the big prize even further. The possibility of jackpot-splitting means the lottery ticket has an expected value less than what it costs you, even when the jackpot tops $300 million. If the jackpot were bigger still, the expected value might tip into the “worth it” zone—or it might not, if the big jackpot attracted an even higher level of ticket sales.* At the time of this account, the biggest Powerball jackpot cited, $588 million, was won by two players, and the biggest lottery jackpot cited in U.S. history, a $688 million Mega Millions prize, was split three ways.

And we haven’t even considered the taxes you’ll pay on your winnings, or the fact that the prize is distributed to you in yearly chunks—if you want all the money up front, you get a substantially smaller payout. And remember, the lottery is a creature of the state, and the state knows a lot about you. In many states, back taxes or other outstanding financial obligations get paid off from lottery winnings before you see a dime. An acquaintance who works at a state lottery told me the story of a man who came to the lottery office with his girlfriend to cash in his $10,000 ticket and spend a wild weekend on the town. When he turned in his ticket, the lottery official on duty told the couple that all but a few hundred dollars of the prize was already committed to delinquent child support the man owed his ex-girlfriend.

This was the first the man’s current girlfriend had heard of the man’s child. The weekend did not go as planned.

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So what’s your best strategy for making money playing Powerball? Here’s my mathematically certified three-point plan:

1. Don’t play Powerball.
2. If you do play Powerball, don’t play Powerball unless the jackpot is really big.
3. And if you buy tickets for a massive jackpot, try to reduce the odds you’ll have to share your haul; pick numbers other players won’t. Don’t pick your birthday. Don’t pick the numbers that won a previous draw. Don’t pick numbers that form a nice pattern on the ticket. And for God’s sake, don’t pick numbers you find in a fortune cookie. (You know they don’t put different numbers in every cookie, right?)

Powerball isn’t the only lottery, but all lotteries have one thing in common; they’re bad bets. A lottery, just as Adam Smith observed, is designed to return a certain proportion of ticket sales to the state; for that to work, the state has to take in more money in tickets than it gives out in prizes. Turning that on its head, lottery players, on average, are spending more money than they win. So players’ expected net return, averaged across the lottery’s ticket sales, is negative. That aggregate loss does not mean every drawing offers the same expected return: a rollover can change the calculation for a particular ticket.