Powerball risk and reward: The true value of a lottery ticket
WASHINGTON, Jan. 13, 2016 — How much would you pay for the chance to win a billion dollars in a lottery?
For a lot of people, $2 is a bargain. California normally sells a million dollars worth of tickets per day; on Saturday, sales were $2.8 million—per hour. Ticket sales reached $277 million on Friday and $400 million on Saturday. Today the Powerball has reached over $1.5 billion after no winning tickets issued in the last two draws.
Tour buses took groups of people to buy tickets. Some sales sites had lines stretching around the block. It’s all about that big potential Powerball jackpot
Forty-four states, the District of Columbia, and two U.S. territories take part in Powerball. Until this drawing, the largest payout for any lottery in North America was $656 million, in a 2012 Mega Millions draw. By Saturday, Powerball was set to pay out nearly $950 million; a winner who took it all in one payment rather than in 30 annual payments would have taken home about $550 million.
That seems worth a $2 ticket, or even a dozen. But what are the odds of winning?
One in 292.2 million. That’s by design of the Multi-State Lottery Association, which revised the lottery to cut the odds from one in 175 million.
Their intent, clearly successful, was to increase the odds of monster jackpots and increase public excitement and ticket sales.
So what’s a ticket worth? First, let’s take out taxes; Uncle Sam gets 40 percent. If you take your money in one lump sum, you take home $330 million after taxes.
That’s nothing to sneeze at, but it’s far from that billion you thought you’d win.
Now let’s look at something called “expected value.” Suppose I toss a coin, and if it comes up heads, I pay you a dollar. If it comes up tails, you pay me a dollar. If it’s a fair coin fair toss, it will come up heads half the time and tails half the time. You have a 50 percent chance of getting a head and receiving a dollar, and a 50 percent chance of getting a tail and paying a dollar.
On any one toss you’ll win or lose, but on average, you come out ahead the same number of times you lose. On average, you get nothing.
Zero is the expected value of that coin-toss game. How much would you pay to play a game that has an expected value of zero? If you’re neutral to risk, you’ll pay zero. If you hate risk, I might have to pay you to play.
If you love risk, you might be willing to pay me to play.
Suppose a lottery offers you a one-in-ten chance of winning $10, and a nine-in-ten chance of winning nothing. If you play that lottery ten time, on average, how many times will you win? Once. So on average, you’ll take home $1, and that’s the expected value of the game. So if you avoid risk, you’ll pay less than a dollar to play it; if you like risk, you might pay more than a dollar.
If you’re neutral to risk, you’ll pay what the game is worth, on average: One dollar.
Your odds of winning the Powerball are one in 292.2 million. So divide that into the payout, and that’s the expected value of the game. Yesterday, the expected value was a bit over a dollar, about $1.10. If we expected multiple winners, the expected value would be even less than that.
Clearly only a crazy person buys a Powerball lottery ticket.
Wednesday’s draw is expected to be well above a billion dollars; most reports now are claiming that it will be $1.3 billion, but given the excitement, that might be low. That means the expected value of a lottery ticket will be higher, but the payout will have to be about $2 billion before the expected value of a lottery ticket reaches the purchase price.
Are all these ticket buyers crazy?
No. Economists and statisticians are crazy when they treat games this way. There’s more to life than expected value.
When you go to a casino, the odds are stacked against you, if only just a little bit. Yet people are happy to pay to play, and they aren’t all risk lovers. Why, then, do they do it? There’s always the thrill of the game: the roll of the dice, the cut of the cards, the spinning numbers on the slot machine.
For some people, it’s entertainment. Who’s to say that the man who pays $500 for a concert ticket is any smarter or more rational than the one who spends (loses) $500 playing the slot machines or poker? It might seem pointless to you, but it’s less pointless in my opinion than listening to Lady Gaga.
One man’s mind-numbing drivel is another man’s entertainment.
Expected value misses more than that. Consider that coin-toss game. Suppose you’re willing to pay nothing to play, and pay nothing not to play. That is, you just don’t care. But let’s change the payouts.
Heads, I pay you $100,000. Tails, you pay me $100,000. Are you still indifferent about playing? The expected value is still zero, but most people would like to avoid that kind of downside risk.
You might pay $50 or $100 or more to avoid the risk of losing $100,000. That’s why people can sell you insurance and extended warranties.
The downside risk of Powerball is tiny. For $2, you get some pleasant anticipation and the chance, however miniscule, of fabulous wealth.
What would you do if you won a billion dollars? That’s an entertaining fantasy, one that a lot of people will pay to share. You might have a better shot at being elected president (you don’t, by the way; statisticians claim that the odds of becoming president are seven times better than winning the Powerball, but they forget that the presidential campaign game is rigged against anyone who isn’t a Clinton or a billionaire), but this fantasy is easier and lower cost.
If someone mocks you for buying a Powerball ticket, tell him to go run for president or stand next to a vending machine.
Then enjoy the game.