Why the lottery really is a tax on ignorance

Jeff Jarvis has been on an anti-lottery rant recently, challenging economists to prove that lotteries are not actually depressing the economy.

Calculate the total amount of income -- income at its most spendable -- drained from the economy; how much spending power did we lose? Then look at where the winnings went; what did it build? Then look at the net income to the government for all this and who paid and how much it cost to generate that income. Then answer the question: Are we better off?

Today, Max Power posted a justification for lotteries. Max correctly notes that the utility of a lottery ticket is as much its entertainment value as its chance of winning big bucks. He does a good job of deconstructing the arguments that lotteries are in "poor taste", and sums up as follows:

Maybe a better objection against government lotteries is that they're such a piss-poor ripoff. You don't see government lotteries in Nevada, because the state is full of casinos, and any casino that offered odds as bad as a state lottery would be out of business in a week. The government-enforced monopoly on gambling is what makes lotteries feasible. But the Jarvis objection to state lotteries would presumably go tenfold for legalized gambling.

I've come to the conclusion that this is indeed the best objection to lotteries, and it's a compelling one at that. I don't think people fully appreciate just how bad the odds are in a state lottery. Frankly, I think your odds would be better in a back-alley craps game. Let me crunch some numbers so you can see what I mean.

In Texas, the main biweekly game is Lotto Texas. As it says on the cited page, just pick six numbers correctly out of 54 - in any order! - and you win the millions.

Well. There are 18,595,558,800 possible ways to choose six numbers from 54. This is simply derived by noting that there are 54 ways of choosing the first number, 53 ways of choosing the second, and so on down to 49 ways of choosing the sixth. Multiply these six numbers together for the result.

Of course, as noted, the order doesn't matter. The arithmetic above considers 1-2-3-4-5-6 and 6-5-4-3-2-1 to be two different combinations whereas Lotto Texas treats them as identical. To remove the duplicates, multiply one through six to count up all the possible orderings (you'll get 720) and divide that out from the total. In the end, there are 25,827,165 winning combinations, so your odds of winning are one in 25,827,165.

A corollary of this is that the jackpot has to be at least that much for your one-dollar ticket to have an expected value equal to its cost. That's a fancy way of saying that the top prize must be that high for the odds to be favorable to you, assuming of course that no one else picks the same numbers as you.

How about the lower prizes? Lotto Texas pays off for picking three, four, or five numbers correctly as well. You can see their payout chart here. Let's compute those odds and compare them to the appropriate prize amounts.

As always, there are the same 18,595,558,800 possible ways to choose six numbers from 54. The goal is to figure out how many ways there are to win. For the five-numbers-right case, we'll start by assuming that the first number chosen is not one of yours and the rest are. There are 48 choices for the "wrong" number (if 6 out of 54 are "right", then 48 out of 54 are "wrong"). After that there are 6, 5, 4, 3, and 2 choices for each of the "right" ones. That's 48 times 720, which is 34,560.

Now notice that there are a total of six ways to order where the wrong number is chosen. What's more, there are always 48 "wrong" choices no matter when it is picked. For instance, if one of your numbers is picked on Ball #1, there are 53 balls remaining but only 5 "right" ones, so there are still 48 "wrong" ones. The upshot of this is that there are six ways to order where the wrong ball is picked, and the total number of ways to pick the five "right" balls is the same each time. That means we can multiply the 34,560 ways by 6 to get 207,360. Divide that by 18,595,558,800 and we see that your odds of hitting five numbers are a pinch better than one in 90,000.

Looking at that payout chart, you'd usually collect between $2000 and $4000 for getting five right. To put that in perspective, that's about like someone offering you even money to roll a twelve at the craps table. If you know anyone willing to take that bet, send 'em my way. I've got some dot-com stocks in my portfolio that need a new home.

It's not much better for the four-number case. There are 15 ways to order where the two wrong numbers are chosen. Therer are 48 x 47 ways to pick the two wrong numbers, and 6 x 5 x 4 x 3 ways to choose the right ones. That's 12,182,400 winning combinations, or odds of one in 1526. With the average payout of a bit more than $100, it's like being given three-to-one odds to roll a twelve in craps. Better than the five-numbers case, but still abysmal.

Finally, the three-number payout. There are 20 ways to order the three wrong numbers, 48 x 47 x 46 ways to pick them, and 6 x 5 x 4 ways to pick the three right numbers. That's 249,062,400 winning combinations, or odds of roughly one in 75. The payout here is a sure $5, so it's on par with the four-number case.

The reason this is such a sucker bet is that it's designed to make money for the state. If you look at the Texas Lottery audited financial statement for 2000 and 2001, you'll see that the payout for all games (including scratch-off games and other, smaller pick'em games) was a bit more than 55 cents on the dollar. By comparison, some Vegas casinos brag that their slots pay out 97 cents on the dollar. Of course, Vegas casinos are in business to make money, too. These high-payout slots are designed to give lots of moderate rewards, and the constant ka-ching of coins falling into their trays is to entice people to come inside to play the real games, where the casino has a bigger edge.

There's an irony here in the belief that state lottery money goes towards education funding, a notion that Max Power dispatches. Of course, if our society were doing a truly sufficient job of educating everyone, far fewer people would be tempted by lotteries. Maybe it's just as well that the lottery revenue goes into the general fund.

So where do I stand on the morality of lotteries? I dislike lotteries and never play them. I don't like the idea of the state separating people from their money in such a tawdry fashion, but who am I to say how people should spend their salaries? I do my best to convince people why they shouldn't play, and the rest is up to them.

Max didn't specifically address Jeff's question about whether lotteries are an especially inefficient means of redistributing wealth. I'm not a Professional Economist, but I'll note that the aforementioned financial statements shows that Texas sold $2.8 billion worth of tickets each of the last two years and cleared about $800 million in revenue after prizes and overhead. We're facing a large budget shortfall this year, due to a slower economy and a property tax cut courtesy of our previous governor. If we had that extra $2 billion per year, it'd make a sizeable dent in the deficit. I daresay those who spend the most on lottery tickets will feel the greatest effect of whatever measures the state implements to acheive its constitutionally-mandated balanced budget. Make of that highly nonscientific observation what you will.

Posted by Charles Kuffner on April 22, 2002 to Technology, science, and math