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# The 10 types of people who buy lottery tickets, explained with math and bad drawings

An illustrated guide.

The Mega Millions jackpot hit \$1.5 billion this week: the second-largest lottery prize in the history of the world. In a flash, office water coolers became math seminars. Ordinary citizens, in broad daylight, could be heard talking combinatorics and expected value.

As a math teacher, I find this thrilling. Imagine a stamp collector’s joy if CNN launched into breathless coverage of a philatelic convention. I spent last year writing a book (Math with Bad Drawings: Illuminating the Ideas That Shape Our Reality) about the intersections between mathematical theory and everyday life, and found myself drawn to the lottery. More than just a probabilistic game, it’s a psychological one. Each year, roughly half of US adults play the lottery, for reasons as diverse as the players themselves.

As we wait to see who collects on the winning ticket purchased in South Carolina, it’s a perfect time to investigate the mathematical and psychological appeal of turning your money into commodified chance. So come, join me, and let’s see who we meet in line for the lottery.

### 1) The Gamer

Behold! It’s the Gamer, who buys lottery tickets for the same reason I buy croissants: not for sustenance but for pleasure.

Take the Massachusetts scratch game entitled \$10,000 Bonus Cash, which costs \$1 to play. On the back, you’ll find the following complicated odds of victory:

What is this ticket worth? Well, we don’t know yet. Maybe \$10,000; maybe \$5; maybe (by which I mean “very probably”) nothing.

It’d be nice to estimate its value with a single number. To get there, let’s imagine that we spent not a mere \$1, but \$1 million on these tickets. In our million tickets, a one-in-a-million event will occur roughly once. A 1-in-100,000 event will occur roughly 10 times. And a 1-in-4 event will occur 250,000 times, give or take.

Sifting through our stacks upon stacks of overstimulating paper, we’d expect the results to look something like this:

About 20 percent of our tickets are winners. Totaling up all of the prizes, our \$1 million investment yields a return of roughly \$700,000 . . . which means that we have poured \$300,000 directly into the coffers of the Massachusetts state government.

Put another way: on average, these \$1 tickets are worth about \$0.70 each.

Mathematicians call this the ticket’s expected value. I find that a funny name, because you shouldn’t “expect” any given ticket to pay out \$0.70, any more than you would “expect” a family to have 1.8 children. I prefer the phrase “long-run average”: It’s what you’d make per ticket if you kept playing this lottery over and over and over and over and over . . .

Sure, it’s \$0.30 less than the price you paid, but entertainment ain’t free, and the Gamer is happy to oblige. Poll Americans about why they buy lottery tickets, and half will say not “for the money” but “for the fun.” These are Gamers. They’re why, when states introduce new lottery games, overall sales rise. Gamers don’t see the new tickets as competing investment opportunities — which would lead to a corresponding drop in sales for old tickets — but as fresh amusements, like extra movies at the multiplex.

What’s the precise attraction for the Gamer? Is it the gratification of victory, the adrenaline rush of uncertainty, the pleasant buzz of watching it all unfold? Well, it depends on each Gamer’s appetites.

I’ll tell you what it isn’t: the financial benefit. In the long run, lottery tickets will almost always cost more than they’re worth.

### 2) The Educated Fool

Wait ... “Almost always”? Why “almost”? What state is dumb enough to sell lottery tickets whose average payout is more than their cost?

These exceptions emerge because of a common rule by which big-jackpot lotteries sweeten the deal: if no one wins the jackpot in a given week, then it “rolls over” to the next week, resulting in an even bigger top prize. Repeat this enough times, and the ticket’s expected value may rise above its price. That’s how we got this week’s Mega Millions blockbuster, where each \$2 ticket had an expected value of over \$5. In fact, had a winning ticket not been purchased, the jackpot would have risen further, to a vertigo-inducing \$2 billion.

In the queue for lotteries like this, you’ll meet a very special player, a treat for gambling anthropologists like us. It’s the Educated Fool, a rare creature who does with “expected value” what the foolish always do with education: mistake partial truth for total wisdom.

“Expected value” distills the multifaceted lottery ticket, with all its prizes and probabilities, down to a one-number summary. That’s a powerful move. It’s also simplistic. The educated fool, by relying exclusively on this one statistic, mistakes lottery tickets for investment opportunities.

Take these two tickets, each costing \$1.

Spend \$10 million on A, and you expect to earn back only \$9 million, amounting to a \$0.10 loss per ticket. Meanwhile, \$10 million spent on B should yield \$11 million in prizes, and thus a profit of \$0.10 per ticket. So to those enamored with expected value, the latter is a golden opportunity, while the former is fool’s gold.

And yet ... would \$11 million bring me any greater happiness than \$9 million? Both values exceed my current bank account many times over. The psychological difference is negligible. So why judge one a rip-off and the other a sweet bargain?

Even simpler, imagine that Bill Gates offers you a wager: For \$1, he’ll give you a one in a billion chance at \$10 billion. Calculating the expected value, you start to salivate: \$1 billion spent on tickets would yield an expected return of \$10 billion. Irresistible!

Even so, Educated Fool, I beg you to resist. You can’t afford this game. Scrape together an impressive \$1 million, and the 99.9 percent likelihood is still that rich man Gates walks away \$1 million richer while you walk away broke. Expected value is a long-run average, and with Gates’s offer, you’ll exhaust your finances well before the “long run” ever arrives.

The same holds true in most lotteries. Perhaps the ultimate repudiation of expected value is the abstract possibility of \$1 tickets like this:

If you buy 10 tickets, you’re likely to win \$1. That’s pretty terrible: only \$0.10 per ticket.

If you buy 100 tickets, you’re likely to win \$20. (That’s 10 of the smallest prize, and one of the next smallest.) Slightly less terrible: now \$0.20 per ticket.

If you buy 1000 tickets, you’re likely to win \$300. (That’s a hundred \$1 prizes, ten \$10 prizes, and a single \$100 prize.) We’re up to \$0.30 per ticket.

Keep going. The more tickets you buy, the better you can expect to do. If you somehow buy a trillion tickets, the likeliest outcome is that you will win \$1.20 per ticket. A quadrillion tickets? Even better: \$1.50 per ticket. In fact, the more tickets you buy, the greater your profit per ticket. If you could somehow invest \$1 googol dollars, you’d be rewarded with \$10 googol in return. With enough tickets, you can attain any average return you desire. The ticket’s expected value is infinite.

But even if you trusted a government to pay out, you could never afford enough tickets to glimpse that profit. Go ahead and spend your life savings on these, Educated Fool. The overwhelming likelihood is bankruptcy.

We humans are short-run creatures. Better leave the long-run average to the immortals.

### 3) The Lackey

Look who just stepped into the queue. It’s the Lackey!

Unlike most folks here, the Lackey’s windfall is guaranteed. That’s because the Lackey isn’t keeping the tickets, just pulling down a wage to stand in line, purchase them, and pass them off to someone else. Modest compensation, but ultra reliable.

Who would pay such a Lackey? And why has the Lackey pretty much gone extinct since the early ’90s? Well, that’s a question for our next player...

### 4) The Big Roller

At first glance, this character looks an awful lot like the Educated Fool: the same scheming grin, the same obsessive focus on expected value. But watch what happens when he encounters a lottery with positive expected value. Whereas the Fool flails, buying a hapless handful of tickets and rarely winning, the Big Roller masterminds a simple and nefarious plan. To transcend risk, you can’t just buy a few tickets. You’ve got to buy them all.

How to become a Big Roller? Just follow these four steps, as elegant as they are loony.

Step No. 1: Seek out lotteries with positive expected value. This isn’t as rare as you’d think. Researchers estimate that 11 percent of lottery drawings fit the bill.

Step No. 2: Beware the possibility of multiple winners. Big jackpots draw more players, increasing the chance that you’d have to split the top prize. That’s an expected-value-diminishing disaster.

Step No. 3: Keep an eye on smaller prizes. In isolation, these consolation prizes (e.g., for matching four out of six numbers) aren’t worth much, but their reliability makes them a valuable hedge. If the jackpot gets split, small prizes can keep the Big Roller from losing too much.

Finally, Step No. 4: When a lottery looks promising, buy every possible combination.

Sounds easy? It’s not. The Big Roller needs extraordinary resources: millions of dollars of capital, hundreds of hours to fill out purchase slips, dozens of Lackeys to complete the purchases, and retail outlets willing to cater to massive orders.

To understand the challenge, witness the dramatic tale of the 1992 Virginia State Lottery.

That February’s drawing offered the perfect storm of circumstances. Rollovers had driven the jackpot up to a record \$27 million. With only 7 million possible number combinations, that gave each \$1 ticket an expected value of nearly \$4. Even better, the risk of a split prize was reassuringly small: Only 6 percent of previous Virginia lotteries had resulted in shared jackpots, and this one had climbed so high that it would yield profit even if shared three ways.

And so the Big Roller pounced. An Australian syndicate of 2,500 investors, led by mathematician Stefan Mandel, made their play. They worked the phones, placing enormous orders at the headquarters of grocery and convenience store chains.

The clock labored against them. It takes time to print tickets. One grocery chain had to refund the syndicate \$600,000 for orders it failed to execute. By the time of the drawing, the investors had acquired only 5 million of the 7 million combinations, leaving a nearly 1 in 3 chance that they would miss out on the jackpot altogether.

Luckily, they didn’t — although it took them a few weeks to surface the ticket from among the stacks of 5 million. After a brief legal waltz with the state lottery commissioner (who vowed “this will not happen again”), they claimed their prize.

Big Rollers like Mandel enjoyed profitable glory days in the 1980s and early 1990s. But those heady times have vanished. As tricky as the Virginia buyout was, it was a logistical breeze compared to the daunting prospect of buying out Mega Millions or Powerball (each with more than 250 million possible combinations). Factor in post-Virginia rules designed to thwart bulk ticket purchases, and it seems the Big Roller may never find the right conditions for a return.

### 5) The Behavioral Economist

For psychologists, economists, probabilists, and various other -ists of the university, nothing is more intriguing than how people reckon with uncertainty. How do they weigh danger against reward? Why do some risks appeal while others repel? But in addressing these questions, researchers encounter a problem: Life is super complicated. Ordering dessert, changing jobs, marrying that good-looking person with the ring — to make these choices is to roll an enormous die with an unknown number of irregular faces. It’s impossible to imagine all the outcomes or to control for all the factors.

Lotteries, by contrast, are simple. Plain outcomes. Clear probabilities. A social scientist’s dream. You see, the behavioral economist is not here to play, just to watch you play.

The scholarly fondness for lotteries goes back centuries. Take the beginnings of probability in the late 1600s. This age saw the birth of finance, with insurance plans and investment opportunities beginning to spread. But the nascent mathematics of uncertainty didn’t know what to make of those complex instruments. Instead, amateur probabilists turned their eye to the lottery, whose simplicity made it the perfect place to hone their theories.

More recently, the wonder duo of Daniel Kahneman and Amos Tversky identified a powerful psychological pattern on display in lotteries. For that, I introduce you to our next line-buddy ...

### 6) The Person with Nothing Left to Lose

In the spirit of behavioral economics, here’s a fun would-you-rather:

In the long run, it doesn’t matter. Choose B 100 times, and (on average) you’ll get 90 windfalls, along with 10 disappointing zeroes. That’s a total of \$90,000 across 100 trials, for an average of \$900. Thus, it shares the same expected value as A.

Yet if you’re like most people, you’ve got a strong preference. You’ll take the guarantee, rather than grab for a little extra and risk leaving empty-handed. Such behavior is called risk-averse.

Now, try this one:

It’s the mirror image of Question No. 1. In the long run, each yields an average loss of \$900. But this time, most people find the guarantee less appealing. They’d rather accept a slight extra penalty in exchange for the chance to walk away scot-free. Here, they are risk-seeking.

These choices are characteristic of prospect theory, a model of human behavior. When it comes to gains, we are risk-averse, preferring to lock in a guaranteed profit. But when it comes to losses, we are risk-seeking, willing to roll the dice for the chance to avoid a bad outcome.

A crucial lesson of prospect theory is that framing matters. Whether you call something a “loss” or a “gain” depends on your current reference point. Take this contrast:

The questions offer identical choices: (a) you walk away with \$1500, or (b) you flip a coin to determine whether you end with \$1000 or \$2000. But folks don’t give identical responses. In the first case, they prefer the \$1500 guaranteed; in the second, they favor the risk. That’s because the questions create different reference points. When the \$2000 is “already yours,” the thought of losing it stresses you out. You’re willing to take risks to avoid that fate.

When life is hard, we roll the dice.

This line of research has a sad and illuminating implication for lottery ticket purchasers. If you’re living under dire financial conditions — if every day feels like a perpetual loss — then you’re more willing to risk money on a lottery ticket.

Think of how a basketball team, trailing late in the game, will begin fouling its opponents. Or how a political candidate, behind with two weeks until the election, will go on the attack, hoping to shake up the campaign. These ploys harm your expected value. You’ll probably lose by an even larger margin than before. But by heightening the randomness, you boost your chances at winning. In times of desperation, that’s all people want.

Researchers find that those who buy lottery tickets “for the money” are far likelier to be poor. For them, the lottery isn’t an amusing way to exercise wealth, but a risky path to acquire it. Yes, on average, you lose. But if you’re losing already, that’s a price you may be willing to pay.

### 7) The Kid that Just Turned 18

Hey, look at that fresh-faced young lottery player! Does the sight fill you with nostalgia for your own youth?

Or does it make you think, “Hey, why are states in the business of selling a product so addictive that they feel compelled to protect minors from buying it?”

Well, there’s someone else I’d like you to meet ...

### 8) The Dutiful Taxpayer

Say hello to our next friend, a would-be civic hero: the Dutiful Taxpayer.

Nobody likes being hit up for money, even by the friendly neighborhood government. States have tried taxing all sorts of things (earnings, retail, real estate, gifts, inheritance, cigarettes), and none of it brings their citizens any special joy. Then, in the 1970s and 1980s, these governments stumbled upon an ancient mind trick.

Turn tax paying into a game, and people will line up down the street to play.

The proposition “I’ll keep \$0.30 from each dollar” prompts grumbles.

Adjust it to “I’ll keep \$0.30 from each dollar and give the entire remainder to one person chosen at random,” and you’ve launched a phenomenon. States don’t offer lotteries out of benevolence; they do it to exploit one of the most effective money-making schemes ever devised. State lotteries deliver a profit of \$30 billion per year, accounting for more than 3% of state budgets.

If you’re picking up a funny odor, I suspect it’s hypocrisy. It’s a bold move to prohibit commercial gambling and then run your own gambling ring out of convenience stores. Heck, the “Daily Numbers” game that states embraced during the 1980s was a deliberate rip-off of a popular illegal game.

Is there an innocent explanation? To be fair, public lotteries are less corruption-prone than private ones. Just ask post-USSR Russians, who found themselves inundated with unregulated mob-run lotteries. Still, if we’re merely aiming to shield gamblers from exploitation, why run aggressive advertising campaigns? Why pay out so little in winnings? And why print gaudy tickets that resemble nightclub fliers spliced with Monster Energy drinks? The explanation is obvious: lotteries are for revenue, period.

To fend off attacks like the one I’m making now, states employ a clever ruse: earmarking the funds for specific popular causes. “We’re running a lottery to raise money” sounds better when you append “for college scholarships” or “for state parks” than it does with “for us—you know, to spend!” This tradition goes way back: Lotteries helped to fund churches in 15th-century Belgium, universities like Harvard and Columbia, and even the Continental Army during the American Revolution.

Earmarks soften the lottery’s image, helping to attract otherwise non-gambling citizens. These are your Dutiful Taxpayers: not necessarily inclined toward games of chance, but willing to play “for a good cause.” Alas, they’re getting hoodwinked. Tax dollars are fungible, and lottery revenue is generally offset, dollar for dollar, by cuts in spending from other sources. As comedian John Oliver quips: designating lottery money for a particular cause is like promising “to piss in one corner of a swimming pool. It’s going all over the place, no matter what you claim.”

So why do earmarks — and state lotteries, for that matter — endure? Because a play-if-you-feel-like-it lottery sounds a lot better than a pay-or-you’ll-go-to-jail tax.

### 9) The Dreamer

Now, allow me to introduce you to a starry-eyed hopeful, who holds as special a place in my heart as the lottery holds in theirs: the Dreamer.

For the Dreamer, a lottery ticket isn’t a chance to win money. It’s a chance to fantasize about winning money. With a lottery ticket in hand, your imagination can go romping through a future of wealth and glory, champagne and caviar, stadium skyboxes and funny-shaped two-seater cars. Never mind that winning a lottery jackpot tends to make people less happy, a trend well documented by psychologists. Daydreaming about that jackpot affords a blissful few minutes while driving your disappointing regular-shaped car.

The prime rule of fantasy is that the top prize must be enough to change your life, transporting you into the next socioeconomic stratum. That’s why instant games, with their modest top prizes, draw in low-income players. If you’re barely scraping together grocery money each week, then \$10,000 carries the promise of financial transformation. By contrast, comfortable middle-class earners prefer games with multimillion-dollar jackpots —enough kindling for a proper daydream.

If you’re seeking an investment opportunity, it’s crazy to fixate on the possible payout while ignoring the probability. But if you’re seeking a license for fantasy, then it makes perfect sense.

### 10) The Enthusiast for Scratching Things

Ah, this citizen knows where it’s at. Forget the cash prizes and all that probability mumbo-jumbo. The chance to rub a quarter across some cardboard is winnings enough. The state lottery: It’s like a scratch-and-sniff for grown-ups.