The story of when humans discovered probability theory. : The Indicator from Planet Money The concept of probability may feel intuitive today, but for much of human history, that wasn't the case. Jacob Goldstein tells the origin story of probability.
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Probability, Gambling, And Death

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Probability, Gambling, And Death

Probability, Gambling, And Death

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Hey, everyone. This is THE INDICATOR FROM PLANET MONEY. I'm Cardiff Garcia.

JACOB GOLDSTEIN, BYLINE: And I'm Jacob Goldstein.

GARCIA: And, Jacob, you, of course, are one of the hosts of our sibling podcast, Planet Money. And you are back on THE INDICATOR now to tell us another one of the stories in your fantastic new book "Money: The True Story Of A Made-Up Thing." And today's topic is the invention of probability theory.

GOLDSTEIN: So probability theory - basically, the study of the uncertain future. But, you know, if you just step back, it is also sort of a way of thinking about the world. And it's a way that feels intuitive to us. You know, if you go look at FiveThirtyEight to see their estimate of the chance that Joe Biden or Donald Trump is going to win the election - as I do many times a day...

GARCIA: (Laughter).

GOLDSTEIN: ...Or if you just look at the weather app on your phone to see what are the chances it's going to rain tomorrow, these are probabilistic predictions.

GARCIA: Yeah, and that does feel natural. Like, obviously, we can try to make these kinds of predictions. But for most of human history, actually, people did not think this way. They thought the future was basically unknowable, that it was controlled by God or the gods or by fate or Mother Nature or whatever.

GOLDSTEIN: And so this idea that has come to feel natural, that we can think about the future in this probabilistic way, is something that people basically invented or at least discovered.

GARCIA: Yeah, and something about this that I kind of love is that we sort of think of probability as this kind of wonky, mathy (ph), very highfalutin but respectable thing. But probability theory actually has its roots not in universities or in higher learning but in gambling and in death.

GOLDSTEIN: So good, let's do both of those.


GOLDSTEIN: Let's do gambling and death. We'll start with gambling because the gamblers came first.

GARCIA: Got to go with the debauchery first, yeah.

GOLDSTEIN: Yeah. Sure, lean in. And so there is, in fact, this one moment, this one exchange of letters we can point to as the moment when probability theory was born. There is this exchange of letters in 1654 between these two French math geniuses, Pierre de Fermat and Blaise Pascal, who, besides being a mathematician, was a big gambler. And they were trying to figure out a problem that gamblers had been thinking about for a few hundred years by this moment. It was called the problem of the points.

GARCIA: And to explain this problem, Jacob, you and I are going to do a simulation. You and I are going to bet on a series of three coin flips. We're going to bet $100.

GOLDSTEIN: OK, big money.

GARCIA: Yeah, best 2 out of 3 coin flips. I've got a quarter here, and I am calling tails. You've got heads. Here comes the flip.


GARCIA: OK. And it came up tails. All right. excellent.

GOLDSTEIN: OK, tails. So you're up 1-0.

GARCIA: Correct.

GOLDSTEIN: But now, you know what? We got to keep going with the show. We can't keep flipping the coin. We're going to stop the game before it's done, OK? So now we've stopped the game before the end, and the question is, how do we split the pot? So this is what Fermat and Pascal were trying to solve, and they figured it out. Their solution was, first, we got to look at every possible outcome of our coin flips and figure out who wins in each one.

GARCIA: We've already had that first flip that came up tails, so I only need one more flip to come up tails to win out of the next two flips. And so there are actually three different combinations where I can win.

GOLDSTEIN: And then for me, I will only win if heads comes up on both of the next two flips. So there's only one combination of the remaining flips where I win. So there's four possible combinations - three where you win and one where I win. You have a 3 out of 4 chance of winning, 75%, I have a 1 in 4 chance of winning, 25%. So Pascal and Fermat decide the fair way to split that pot, that $100 that we bet, is $75 for you and $25 for me. That is the solution to the problem of the points.

GARCIA: Yeah. And I got to say it seems not at all surprising, like, just kind of an obvious and simple math problem. Like, that was not hard.

GOLDSTEIN: A little bit of a letdown, right?

GARCIA: Slightly.

GOLDSTEIN: Little bit of a letdown. I - actually, though, I actually think the obviousness, to us, of this problem is sort of the point here, right? Like, before these guys figured this out, this was just not the way people thought about the world. You know, they thought of the future as unknowable not as something you could calculate. And because they figured it out and because probability spread from gambling to so many other parts of the world, it has become sort of second nature to us. It has become obvious in this way.

GARCIA: Yeah. And also, I can see the appeal of thinking that the gods had fated me to win this coin flip instead of it just being probability.

GOLDSTEIN: Yeah, and me too; it's the gods' fault that I was lost.

GARCIA: OK. So after a quick break, we are then going to talk about how probability moved on from gambling, from debauchery, to something a little bit more morbid - to death.

GOLDSTEIN: So the guy who gets us - gets probability theory into death is Edmond Halley. This is the 1690s now, a few decades later. Halley is British. He's already, by this point, helped his pal Isaac Newton publish "The Principia." He has not yet figured out the comet thing. That's coming.

GARCIA: Halley's Comet.

GOLDSTEIN: This is the Halley.


GOLDSTEIN: So the problem Halley is actually trying to solve at this moment is how to price annuities.

GARCIA: Womp, womp, womp (ph) - not a comet, annuities we're talking about here. OK fine.

GOLDSTEIN: You mean dada-da-da-dada (ph) - annuities.

GARCIA: Yeah (laughter). So an annuity is a kind of financial product. It still exists today. And here's how it works. When I buy an annuity, I pay a lump sum of money up front - let's say $100,000. And then in exchange, I get a payment every year until I die, which might be, say, $5,000 a year. And obviously, whether or not this is a good deal depends entirely on how long I live. Like, if I buy an annuity and then drop dead the very next day, that is a horrible deal for me. I gave up $100,000, and I didn't get anything back. But if I buy an annuity and I live for another 60 years, that is a pretty sweet deal.

GOLDSTEIN: So Halley was interested in annuities because the British government was big in the annuities business at the time. The British government sold annuities. But here is the sort of wild part. The rates on the annuities were the same no matter the age of the buyer. So if you were 20 years old and likely to live for decades or 80 years old and not likely to live for decades, you got the same deal from the government.

So what Halley does is he gets this record of births and deaths from this one town that just happens to keep really good records of these things. And then he does a bunch of math. He uses this new, you know, probability theory, and he publishes a paper laying out how likely people are to die in a given span of time. And he finds, you know, somebody who just turned 20 has a 1% chance of dying in the next year. A 50-year-old has a 3% chance of dying in the next year. He even does some weirder ones. Like, here's a quote; "a man of 30 may reasonably expect to live between 27 and 28 years." So he lays out all these life expectancies, and then he turns to annuities. He says the government should sell annuities at a price where the buyer gets back the money they pay upfront if they live for the average amount of years - reasonable enough. So, you know, a 20-year-old should get paid less each year than an 80-year-old.

GARCIA: So I take it the government then did exactly that, right?

GOLDSTEIN: They did that, but it took them, like, a hundred years.


GOLDSTEIN: For decades and decades, they kept selling annuities at a fixed price regardless of age. And I think this speaks to, really, just how different this probabilistic way of thinking was. You know, it seems so obvious to us, in part because since Halley's time, probability has spread to so many different things. You know, it's in finance. It's in medicine. It's in insurance. It's in weather apps and election predictions. But when it happened, it was really this giant intellectual leap in human thought, and so it took a while to sink in.

GARCIA: Jacob Goldstein, thanks so much, man.

GOLDSTEIN: Thanks very much, man. The book is "Money: The True Story Of A Made-Up Thing."

GARCIA: This episode of THE INDICATOR was produced by Darian Woods and fact-checked by Sean Saldana. Our editor is Paddy Hirsch, and THE INDICATOR is a production of NPR.


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