How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843)
Page 20
The notion of expected value started to come into mathematical focus in the mid-1600s, and by the end of that century, the idea was understood well enough to be used by practical scientists like Edmond Halley, the Astronomer Royal of England.* Yep, the comet guy! But he was also one of the first scientists to study the correct pricing of insurance, which in the reign of William III was a matter of critical military importance. England had thrown itself enthusiastically into war on the continent, and war required capital. Parliament proposed to raise the necessary funds via the “Million Act” of 1692, which aimed to raise a million pounds by selling life annuities to the population. Signing up for an annuity meant paying the Crown a lump sum, in exchange for a guaranteed lifetime annual payout. This is a kind of life insurance in reverse; purchasers of such an annuity are essentially betting that they won’t die anytime soon. As a measure of the rudimentary state of the actuarial science of the time, the cost of the annuity was set without reference to the annuitant’s age!* A lifetime annuity for a grandfather, likely to require funding for at most a decade, cost as much as one for a child.
Halley was scientist enough to understand the absurdity of the age-independent pricing scheme. He determined to work out a more rational accounting of the value of a lifetime annuity. The difficulty is that people don’t arrive and depart on a rigid schedule, as comets do. But by using birth and death statistics, Halley was able to estimate the probability of various life spans for each annuitant, and thereby to compute the expected value of the annuity: “It is plain that the purchaser ought to pay for only such a part of the value of the annuity, as he has chances that he is living; and this ought to be computed yearly, and the sum of all those yearly values being added together, will amount to the value of the annuity for the life of the person proposed.”
In other words: Grandpa, with his shorter expected life span, pays less for an annuity than Junior.
“EET EES OBVIOUS.”
Digression: when I tell people the story of Edmond Halley and the price of annuities, I often get interrupted: “But it’s obvious that you should charge younger people more!”
It is not obvious. Rather, it is obvious if you already know it, as modern people do. But the fact that people who administered annuities failed to make this observation, again and again, is proof that it’s not actually obvious. Mathematics is filled with ideas that seem obvious now—that negative quantities can be added and subtracted, that you can usefully represent points in a plane by pairs of numbers, that probabilities of uncertain events can be mathematically described and manipulated—but are in fact not obvious at all. If they were, they would not have arrived so late in the history of human thought.
This reminds me of an old story from the Harvard math department, concerning one of the grand old Russian professors, whom we shall call O. Professor O is midway through an intricate algebraic derivation when a student in the back row raises his hand.
“Professor O, I didn’t follow that last step. Why do those two operators commute?”
The professor raises his eyebrows and says, “Eet ees obvious.”
But the student persists: “I’m sorry, Professor O, I really don’t see it.”
So Professor O goes back to the board and adds a few lines of explanation. “What we must do? Well, the two operators are both diagonalized by . . . well, it is not exactly diagonalized but . . . just a moment . . .” Professor O pauses for a little while, peering at what’s on the board and scratching his chin. Then he retreats to his office. About ten minutes go by. The students are about to start leaving when Professor O returns, and again assumes his station in front of the chalkboard.
“Yes,” he says, satisfied. “Eet ees obvious.”
DON’T PLAY POWERBALL
The nationwide lottery game Powerball is currently playable in forty-two U.S. states, the District of Columbia, and the U.S. Virgin Islands. It’s 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 an $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 today, 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 two dollars 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 migh
t 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 want to 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
or 99.9999994%. 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 that happens, your share of the $337 million prize drops to a puny $168 million. That cuts the expected value of the jackpot to
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.* The biggest Powerball jackpot yet, $588 million, was won by two players, and the biggest lottery jackpot 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.
—
So what’s your best strategy for making money playing Powerball? Here’s my mathematically certified three-point plan:
Don’t play Powerball.
If you do play Powerball, don’t play Powerball unless the jackpot is really big.
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 the expected value of a lottery ticket has to be negative.
Except when it’s not.
THE LOTTERY SCAM THAT WASN’T
On July 12, 2005, the Compliance Unit of the Massachusetts State Lottery received an unusual phone call from an employee at a Star Market in Cambridge, the northern suburb of Boston that houses both Harvard and MIT. A college student had come into the supermarket to buy tickets for the state’s new Cash WinFall game. That wasn’t strange. What was unusual was the size of the order; the student had presented fourteen thousand order slips, each one filled out by hand, for a total of $28,000 in lottery tickets.
No problem, the lottery told the store; if the slips are filled out properly, anybody can play as much as they want. Stores were required to get a waiver from the lottery office if they wanted to sell more than $5,000 in tickets per day, but those waivers were easily granted.
That was a good thing, because the Star wasn’t the only Boston-area lottery agent doing a vigorous business that week. Twelve more stores contacted the lottery in advance of the July 14 drawing to ask for waivers. Three of those were concentrated in a heavily Asian-American neighborhood of Quincy, just south of Boston on the bay shore. Tens of thousands of Cash WinFall tickets were being sold to a small group of buyers at a handful of stores.
What was going on? The answer wasn’t secret; it was in plain sight, right there in the rules for Cash WinFall. The new game, launched in the fall of 2004, was a replacement for Mass Millions, which had been phased out after going an entire year without paying out a jackpot. Players were getting discouraged, and sales were down. Massachusetts needed to shake up its lottery, and state officials hit on the idea of adapting WinFall, a game from Michigan. In Cash WinFall, the jackpot didn’t pile higher and higher with each week it went unclaimed; instead, every time the pot went over $2 million, the money “rolled down” to enhance the lesser prizes that weren’t so hard to win. The jackpot reset to its minimum $500,000 value for the following drawing. The lottery commission hoped the new game, which made it possible to take in serious winnings without hitting the jackpot, would seem like a good deal.
They did their job too well. In Cash WinFall, Massachusetts had inadvertently designed a game that actually was a good deal. And by the summer of 2005, a few enterprising players had figured that out.
On a normal day, here’s how the prize distribution for Cash WinFall looked:
match all 6 numbers
1 in 9.3 million
variable jackpot
match 5 of 6
1 in 39,000
$4,000
match 4 of 6
1 in 800
$150
match 3 of 6
1 in 47
$5
match 2 of 6
1 in 6.8
free lottery tic
ket
If the jackpot is $1 million, the expected value of a two-dollar ticket is pretty poor:
($1 million / 9.3 million) + ($4,000 / 39,000) + ($150 / 800) + ($5 / 47) + ($2 / 6.8) = 79.8 cents.
That’s a rate of return so pathetic it makes Powerball players look like canny investors. (And we’ve generously valued a free ticket at the $2 it would cost you instead of the substantially smaller expected value it brings you.)
But on a roll-down day, things look very different. On February 7, 2005, the jackpot stood near $3 million. Nobody won that jackpot—unsurprising, considering that only about 470,000 people played Cash WinFall that day, and matching all six numbers was about a 1-in-10 million long shot.
So all that money rolled down. The state’s formula rolled $600,000 to the match-5 and match-3 prize pools and $1.4m into the match-4s. The probability of getting four out of six WinFall numbers right is about 1 in 800, so there must have been about six hundred match-4 winners that day out of the 470,000 players. That’s a lot of winners, but $1.4 million dollars is a lot of money. Dividing it into six hundred pieces leaves more than $2,000 for each match-4 winner. In fact, you’d expect the payout for matching 4 out of 6 numbers that day to be around $2,385. That’s a much more attractive proposition than the measly $150 you’d win on a normal day. A 1-in-800 chance of a $2,385 payoff has an expected value of
$2364 / 800 = $2.98
In other words, the match-4 prize alone makes the ticket worth its two-dollar price. Throw in the other prizes, and the story gets even sweeter.
Prize
Chance of winning
Expected number of winners
Roll-down allocation
Roll-down per prize