It is, of course, a classic reductio ad unlikely:
If there’s no God, it would be unlikely for things as complex as human beings to have developed;
Humans have developed;
Therefore, it’s unlikely there’s no God.
This is much like the argument that the Bible coders used; if God didn’t write the Torah, it’s unlikely that the text on the scroll would so faithfully record the birthdays of the rebbes!
You may be sick of hearing me say it by now, but reductio ad unlikely doesn’t always work. If we really mean to compute in numerical terms how confident we should be that God created the universe, we’d better draw another Bayes box.
The first difficulty is to understand the priors. This is a hard thing to get your head around. For the roulette wheels, we were asking: How likely do we think it is that the wheel is rigged, before we see any of the spins? Now we’re asking: How likely would we think it was that there was a God, if we didn’t know that the universe, the Earth, or we ourselves exist?
At this point, the usual move is to throw up one’s hands and invoke the charmingly named principle of indifference—since there can be no principled way to pretend we don’t know we exist, we just divvy up the prior probability evenly, 50% for GOD and 50% for NO GOD.
If NO GOD is true, then complex beings like humans must have arisen by pure chance, perhaps spurred along by natural selection. Designists then and now agree that this is phenomenally unlikely; let’s make up numbers and say it was a one-in-a-billion-billion shot. So what goes in the bottom right box is one-billion-billionth of 50%, or one in two billion billion.
What if GOD is true? Well, there are lots of ways God could be; we don’t know in advance that a God who made the universe would care to create human beings, or any thinking entities at all, but certainly any God worth the name would have the ability to whip up intelligent life. Perhaps if there’s a God there’s a one in a million chance God would make creatures like us.
So the box now looks like this:
At this point we can examine the evidence, which is that we exist. So the truth lies somewhere in the bottom row. And in the bottom row, you can plainly see that there is a lot more probability—a trillion times more!—in the GOD box than in the NO GOD box.
This, in essence, is Paley’s case, the “argument by design,” as a modern Bayesian type would express it. There are many solid objections to the argument by design, and there are also two billion billion fighty books on the topic of “you should totally be a cool atheist like me” where you can read those arguments, so let me stick here to the one that’s closest to the math at hand: the “cleanest man in school” objection.
You probably know what Sherlock Holmes had to say about inference, the most famous thing he ever said that wasn’t “Elementary!”:
“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth.”
Doesn’t that sound cool, reasonable, indisputable?
But it doesn’t tell the whole story. What Sherlock Holmes should have said was:
“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth, unless the truth is a hypothesis it didn’t occur to you to consider.”
Less pithy, more correct. The people who inferred that my friend was the dirtiest man in school were considering only two hypotheses:
CLEAN: my friend was rotating through his shirts, washing them, then starting the rotation over, like a normal person
DIRTY: my friend was a filthy savage who wore dirty clothes.
You may start with some prior; based on my memory of college, assigning a probability of 10% to DIRTY is about right. But it doesn’t really matter what your prior is: CLEAN is ruled out by the observation that my friend wears the same shirt every day. “When you have excluded the impossible . . .”
But hold up, Holmes—the true explanation, LAZY ENTREPRENEUR, was a hypothesis not on the list.
The argument by design suffers from much the same problem. If the only two hypotheses you admit are NO GOD and GOD, the rich structure of the living world might well be taken as evidence in favor of the latter against the former.
But there are other possibilities. What about GODS, where the world was put together in a hurry by a squabbling committee? Many distinguished civilizations have believed as much. And you can’t deny that there are aspects of the natural world—I’m thinking pandas here—that seem more likely to have resulted from grudging bureaucratic compromise than from the mind of an all-knowing deity with total creative control. If we start by assigning the same prior probability to GOD and GODS—and why not, if we’re going with the principle of indifference?—then Bayesian inference should lead us to believe in GODS much more than GOD.*
Why stop there? There’s no end to the making of origin stories. Another theory with some adherents is SIMS, where we’re not actually people at all, but simulations running on an ultracomputer built by other people.* That sounds bizarre, but plenty of people take the idea seriously (most famously, the Oxford philosopher Nick Bostrom), and on Bayesian grounds, it’s hard to see why you shouldn’t. People like to build simulations of real-world events; surely, if the human race doesn’t extinguish itself, our power to simulate will only increase, and it doesn’t seem crazy to imagine that those simulations might one day include conscious entities that believed themselves to be people.
If SIMS is true, and the universe is a simulation constructed by people in a realer world, then it’s pretty likely there’d be people in the universe, because people are people’s favorite things to simulate! I’d call it a near certainty (for the sake of the example, let’s say an absolute certainty) that a simulated world created by technologically advanced humans would have (simulated) humans in it.
If we assign each of the four hypotheses we’ve met so far a prior probability of 1/4, the box looks something like this:
Given that we actually do exist, so that the truth is in the bottom row, almost all the probability is sitting in SIMS. Yes, the existence of human life is evidence for the existence of God; but it’s much better evidence that our world was programmed by people much smarter than us.
Advocates of “scientific creationism” hold that we should argue in the classroom for the existence of a world-designer, not because the Bible says so—that would be unconstitutionally naughty!—but on coolly reasonable grounds, founded on the astonishing unlikelihood of the existence of humanity under the NO GOD hypothesis.
But if we took this approach seriously, we would tell our tenth graders something like this: “Some have argued that it’s highly unlikely for something as complex as the Earth’s biosphere to have arisen purely by natural selection without any intervention from outside. By far the most likely such explanation is that we are actually not physical beings at all, but residents of a computer simulation being carried out by humans with unthinkably advanced technology, to what purpose we can’t exactly know. It’s also possible that we were created by a community of gods, something like those worshiped by the ancient Greeks. There are even some people who believe that one single God created the universe, but that hypothesis should be considered less strongly supported than the alternatives.”
Think the school board would go for this?
I had better hasten to point out that I don’t actually think this constitutes a good argument that we’re all sims, any more than I think Paley’s argument is a good one for the existence of the deity. Rather, I take the queasy feeling these arguments generate as an indication that we’ve reached the limits of quantitative reasoning. It’s customary to express our uncertainty about something as a number. Sometimes it even makes sense to do so. When the meteorologist on the nightly news says, “There’s a 20% chance of rain tomorrow,” what he means is that, among some large population of past days with
conditions similar to those currently obtaining, 20% of them were followed by rainy days. But what can we mean when we say, “There’s a 20% chance that God created the universe?” It can’t be that one in five universes was made by God and the rest popped up on their own. The truth is, I’ve never seen a method I find satisfying for assigning numbers to our uncertainty about ultimate questions of this kind. As much as I love numbers, I think people ought to stick to “I don’t believe in God,” or “I do believe in God,” or just “I’m not sure.” And as much as I love Bayesian inference, I think people are probably best off arriving at their faith, or discarding it, in a non-quantitative way. On this matter, math is silent.
If you don’t buy it from me, take it from Blaise Pascal, the seventeenth-century mathematician and philosopher who wrote in his Pensées, “‘God is, or He is not.’ But to which side shall we incline? Reason can decide nothing here.”
This is not quite all Pascal had to say on the subject. We return to his thoughts in the next chapter. But first, the lottery.
Includes: MIT kids game the Massachusetts State Lottery, how Voltaire got rich, the geometry of Florentine painting, transmissions that correct themselves, the difference between Greg Mankiw and Fran Lebowitz, “I’m sorry, was that bofoc or bofog?,” parlor games of eighteenth-century France, where parallel lines meet, the other reason Daniel Ellsberg is famous, why you should be missing more planes
ELEVEN
WHAT TO EXPECT WHEN YOU’RE EXPECTING TO WIN THE LOTTERY
Should you play the lottery?
It’s generally considered canny to say no. The old saying tells us lotteries are a “tax on the stupid,” providing government revenue at the expense of people misguided enough to buy tickets. And if you see the lottery as a tax, you can see why lotteries are so popular with state treasuries. How many other taxes will people line up at convenience stores to pay?
The attraction of lotteries is no novelty. The practice dates back to seventeenth-century Genoa, where it seems to have evolved by accident from the electoral system. Every six months, two of the city’s governatori were drawn from the members of the Petty Council. Rather than hold an election, Genoa carried out the election by lot, drawing two slips from a pile containing the names of all 120 councilors. Before long, the city’s gamblers began to place extravagant side bets on the election outcome. The bets became so popular that gamblers started to chafe at having to wait until Election Day for their enjoyable game of chance; and they quickly realized that if they wanted to bet on paper slips drawn from a pile, there was no need for an election at all. Numbers replaced names of politicians, and by 1700 Genoa was running a lottery that would look very familiar to modern Powerball players. Bettors tried to guess five randomly drawn numbers, with a bigger payoff the more numbers a player matched.
Lotteries quickly spread throughout Europe, and from there to North America. During the Revolutionary War, both the Continental Congress and the governments of the states established lotteries to fund the fight against the British. Harvard, back in the days before it enjoyed a nine-figure endowment, ran lotteries in 1794 and 1810 to fund two new college buildings. (They’re still used as dorms for first-year students today.)
Not everyone applauded this development. Moralists thought, not wrongly, that lotteries amounted to gambling. Adam Smith, too, was a lottery naysayer. In The Wealth of Nations, he wrote:
That the chance of gain is naturally overvalued, we may learn from the universal success of lotteries. The world neither ever saw, nor ever will see, a perfectly fair lottery, or one in which the whole gain compensated the whole loss; because the undertaker could make nothing by it. . . . In a lottery in which no prize exceeded twenty pounds, though in other respects it approached much nearer to a perfectly fair one than the common state lotteries, there would not be the same demand for tickets. In order to have a better chance for some of the great prizes, some people purchase several tickets; and others, small shares in a still greater number. There is not, however, a more certain proposition in mathematics, than that the more tickets you adventure upon, the more likely you are to be a loser. Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty.
The vigor of Smith’s writing and his admirable insistence on quantitative considerations shouldn’t blind you to the fact that his conclusion is not, strictly speaking, correct. Most lottery players would say buying two tickets instead of one doesn’t make you more likely to be a loser, but twice as likely to be a winner. And that’s right! In a lottery with a simple prize structure, it’s easy to check for yourself. Suppose the lottery has 10 million combinations of numbers and just one is a winner. Tickets cost $1 and the jackpot is $6 million.
The person who buys every single ticket spends $10 million and gets a $6 million prize; in other words, just as Smith says, this strategy is a certain loser, to the tune of $4 million. The small-time operator who buys a single ticket is better off—at least she has a 1 in 10 million chance of coming out ahead!
But what if you buy two tickets? Then your chance of losing shrinks, though admittedly only from 9,999,999 in 10 million to 9,999,998 in 10 million. Keep buying tickets and your chance of being a loser keeps going down, until the point where you’ve purchased 6 million tickets. In that case, your chance of winning the jackpot, and thus breaking even, is a solid 60%, and there’s only a 40% chance of you ending up a loser. Contrary to Smith’s claim, you’ve made yourself less likely to lose money by buying more tickets.
Purchase one more ticket, though, and you’re sure to lose money (though whether it’s $1 or $4,000,001 depends on whether you hold the winning ticket).
It’s hard to reconstruct Smith’s reasoning here, but he may have been a victim of the all-curves-are-lines fallacy, reasoning that if buying all the tickets made you certain to lose money, then buying more tickets must make you more likely to lose money.
Buying 6 million tickets minimizes the chance of losing money, but that doesn’t mean it’s the right play; it matters how much money you lose. The one-ticket player suffers a near certainty of losing money; but she knows she won’t lose a lot. The buyer of 6 million tickets, despite the lower chance of losing, is in a much more dangerous position. And probably you still feel that neither choice seems very wise. As Smith points out, if the lottery is a winning proposition for the state, it seems like it must be a bad idea for whoever takes the other side of the bet.
What Smith’s argument against lotteries is missing is the notion of expected value, the mathematical formalism that captures the intuition Smith is trying to express. It works like this. Suppose we possess an item whose monetary worth is uncertain—like, say, a lottery ticket:
9,999,999/10,000,000 times: ticket is worth nothing
1/10,000,000 times: ticket is worth $6 million
Despite our uncertainty, we still might want to assign the ticket a definite value. Why? Well, what if a guy comes around offering to pay $1.20 for people’s tickets? Is it wise to make the deal and pocket the 20-cent profit, or should I hold on to my ticket? That depends whether I’ve assigned the ticket a worth of more or less than $1.20.
Here’s how you compute the expected value of a lottery ticket. For each possible outcome, you multiply the chance of that outcome by the ticket’s value given that outcome. In this simplified case, there are only two outcomes: you lose, or you win. So you get
9,999,999/10,000,000 × $0 = $0
1/10,000,000 × $6,000,000 = $0.60
Then you add the results up:
$0 + $0.60 = $0.60.
So the expected value of your ticket is 60 cents. If a lottophile comes to your door and offers $1.20 for your ticket, expected value says you ought to make the deal. In fact, expected value says you shouldn’t have paid a dollar for it in the first place!
EXPECTED VALUE IS NOT THE VALUE YOU EXPECT
Expected value is another one of those mathematical notions saddled, like significance, with a name that doesn’t quite capture its meaning. We certainly don’t “expect” the lottery ticket to be worth 60 cents: on the contrary, it’s either worth 10 million clams or zilch, nothing in between.
Similarly: suppose I make a $10 bet on a dog I think has a 10% chance of winning its race. If the dog wins, I get $100; if the dog loses, I get nothing. The expected value of the bet is then
(10% × $100) + (90% × $0) = $10.
But this is not, of course, what I expect to happen. Winning $10 is, in fact, not even a possible outcome of my bet, let alone the expected one. A better name might be “average value”—for what the expected value of the bet really measures is what I’d expect to happen if I made many such bets on many such dogs. Let’s say I laid down a thousand $10 bets like that. I’d probably win about a hundred of them (the Law of Large Numbers again!) and make $100 each time, totaling $10,000; so my thousand bets are returning, on average, $10 per bet. In the long run, you’re likely to come out about even.
Expected value is a great way to figure out the right price of an object, like a gamble on a dog, whose true value isn’t certain. If I pay $12 apiece for those tickets, I’m very likely to lose money in the long run; if I can get them for $8, on the other hand, I should probably buy as many as I can.* Hardly anybody plays the dogs anymore, but the machinery of expected value is the same whether you’re pricing race tickets, stock options, lottery tickets, or life insurance.
THE MILLION ACT
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 19
