Posts Tagged ‘error’

Flat Priors and Other Improbable Tales

September 8, 2010

Some collected and invented stories about erroneous thinking in probability.

A Visit

It’s night. You are coming downstairs for a glass of water. You hear a creaking sound and look around a corner to see a man in a ski mask opening your front door. “What are the odds?” you think. “Normally that guy would have set off my burglar alarm and been scared off by the loud wailing, but he happened to stop by for a visit just one minute after the power went out.”

Feynman

You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won’t believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!

Immortality

About 100 billion people have ever lived, and there are about 7 billion people alive now. Therefore about 7% of people are extremely long-lived.

Astronaut

A man makes it through a long battery of physical and psychological tests and finally achieves his lifelong dream of joining the astronaut program. He immediately takes up heavy smoking. “What gives,” asks his friend. “I thought you were a health nut.”

“I am,” replies the man. “Anybody who smokes a lot will probably die of lung cancer.”

“Why would you want to die of lung cancer?” his friend asks.

“A shuttle explosion will kill you in two seconds,” he replies. “But now I’m gonna die of lung cancer, and that’ll take at least forty years.”

Hitchhiker’s Guide

It is known that there is an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the universe can be said to be zero. From this it follows that the population of the universe is also zero, and that any people you may meet from time to time are merely the product of a deranged imagination.
-Douglas Adams

Foreign Lands

In a certain country, people always name their first child a name that starts with “A”, their second child a name that starts with “B”, etc. Families in this country are anywhere from one to ten children; equal numbers of families have each. It is a tradition in this country for each father to randomly select one of his children each day to accompany him on a walk.

When visiting this country, you meet a man out walking with his daughter, who he introduces as Evelyn. You now know the man has at least five children. If he had exactly five, your chances of meeting the one whose names starts with “E” are 1/5. If he had more, say eight, your chances of meeting the one whose name starts with “E” are 1/8. Therefore, you conclude that it is most likely that Evelyn is the oldest child. You realize there was nothing special about Evelyn, and conclude that any time you meet a man walking with his child in this country, he is most likely to be walking with the oldest one.

Casablanca

Of all the gin joints, in all the towns, in all the world, she walks into mine.

DNA

While rummaging around in his parents’ attic, Sean comes across an old love letter to his mother. It’s from Rodrigo Valenzuela, a man he never knew, to his mother. It refers to “nights of fevered frenzy and mornings of muted passion”, is signed, “a mi amor”, and asks when her husband will be away again. The letter is dated eight months before Sean’s birth date. He looks in the mirror, wondering why he didn’t inherit his parent’s fiery orange hair and why salsa music has always stirred his soul.

Sean looks up information on paternity tests, and finds that if you send in one sample of DNA as the suspected father and one as the suspected child, the test will report a probability, which represents the probability that a man with the “father” DNA would sire a child at least as genetically different from him as the “child” DNA. Thus, a low percentage, like 0.001%, means that a true child would have only small chance of being as different from the father as the “child” sample is. This is the result we expect if the child is not from the father. A high percentage, like 60%, means the “child” and “father” DNA are very close, and is what we expect if the man is the true father.

Secretly, Sean collects a sample of the DNA of the man he’s always called “dad” and one of his own and sends them in for testing. As a control, Sean also collects a sample from his own son, and a second sample from himself and sends this sample in as well. Finally, Sean hunts down Rodrigo Valenzuela using Facebook, “friends” him, studies his “likes” and “interests”, uses them to befriend Roderigo in real life, asks to borrow his car, and steals a hair from the headrest. He sends in a third sample of Rodrigo and himself for testing.

Two weeks later the test results come back. Sean isn’t shocked. The probability for him and his “dad” is a scant 0.00004%. The probability from Roderigo and himself is 7%. Finally, the result from his son is 74%. Sean realizes that there’s some natural variation in the test, but the evidence is still clear: Roderigo is his true father.

The next day the clinic and says there’s been a mix up. They accidentally switched the samples from Sean and his son, so the 74% was actually the result of testing Sean’s son in the “father” role and Sean in the “child”. Sean is understandably upset. He goes to bed that night thinking that although Roderigo may be his father, it’s ten times as likely that his own son will, in the course of his life, discover time travel and go back to impregnate Sean’s mother.

Flat Prior

On whether or not the Large Hadron Collider would create a black hole that would consume Earth:

John Oliver: So, roughly speaking, what are the chances that the world is going to be destroyed? Is it one in a million, one in a billion?

Walter Wagner: Well, the best we can say right now is about a one in two chance.

JO: Hold on a second. Is the, if, 50 – 50?

WW: Yeah, 50-50.
….
WW: It’s a chance. It’s a 50-50 chance.

JO: You keep coming back to this 50-50 thing. It’s weird, Walter.

WW: Well, if you have something that can happen and something that won’t necessarily happen. It’s either gonna happen or it’s gonna not happen. And, so it’s, the best guess is one in two.

JO: I’m not sure that’s how probability works, Walter.

from The Daily Show

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Bounce, Part 6

January 11, 2010

Last time, we looked at what Galileo had to say about free fall. This time, we’ll take one more example from his dialog and try to squeeze a little moral out of it.

Galileo presents his ideas through the character Salviati, who explains them to his companions Sagredo and Simplicio. Salviati’s interlocutors raise all manner of objection to his theories, but Salviati answers them and convinces everyone of his point all the more surely in the process. One such objection is given by Sagredo, who doesn’t believe that the velocity of a falling object increases evenly with each second of falling:

So far as I see at present, the definition might have been put a little more clearly perhaps without changing the fundamental idea, namely, uniformly accelerated motion is such that its speed increases in proportion to the space traversed; so that, for example, the speed acquired by a body in falling four cubits would be double that acquired in falling two cubits and this latter speed would be double that acquired in the first cubit.

Sagredo is suggesting that rather than Galileo’s law

v \propto t,

that the velocity of a falling body increases the same amount each second, we should instead have

v \propto x,

the the velocity increases the same amount each meter the body falls. These are different hypotheses, and so we need to distinguish between them. Given that Salviati states he is not interested in examining the fundamental cause of gravity, and only in characterizing its behavior, there is only one way to do this – experiment.

Instead, Salviati offers the following retort:

…that motion should be completed instantaneously; and here is a very clear demonstration of it. If the velocities are in proportion to the spaces traversed, or to be traversed, then these spaces are traversed in equal intervals of time; if, therefore, the velocity with which the falling body traverses a space of eight feet were double that with which it covered the first four feet (just as the one distance is double the other) then the time-intervals required for these passages would be equal. But for one and the same body to fall eight feet and four feet in the same time is possible only in the case of instantaneous [discontinuous] motion;

What a strange counterargument! It makes absolutely no sense. Gaining an even increment of speed for each unit of time is a perfectly consistent mathematical law, and does not at all imply instantaneous motion. We can write this law as

\frac{dx}{dt} = c(x - x_0),

which implies that the distance fallen increases exponentially with time. This is completely contrary to observation, and it would be hard to build a unified mechanics like Newton’s that respects this law, but it isn’t logically impossible for the reasons Salviati cites.

And how do Salviati’s friends respond to this argument? Do they rip it apart, or restate their objection more clearly, or request further detail?

Sagredo replies,

You present these recondite matters with too much evidence and ease; this great facility makes them less appreciated than they would be had they been presented in a more abstruse manner. For, in my opinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion.

I guess it’s easier to convince the people you’re arguing with when they’re fictional characters you invented yourself!

Everyone makes mistakes, and they hadn’t gotten around to inventing peer review in the sixteenth century, so let’s forgive Galileo, and take a further look at this hypothesis.

Suppose we have a projectile with constant horizontal velocity and vertical velocity that changes according to the distance traveled in the vertical direction.

What happens if we shoot it up out of a cannon? We know from experience that the cannonball slows down, so it must be losing a constant amount of velocity for each unit height it gains.

The cannonball slows down its vertical velocity, but as it does so, its vertical position changes less. Since its vertical position changes less, the change in its vertical velocity slows down more. In fact, the cannonball asymptotically reaches a certain height above ground, and then stays there!

If the vertical height isn’t changing, then according to this law the vertical distance traveled is zero, and because vertical velocity only changes when vertical height changes, the vertical velocity stays zero. This cannonball would never come down.

On the other hand, if it were pushed down just a little bit, it would gain speed very rapidly, falling exponentially back toward Earth. The motion under Sagredo’s law is absurd, but I wonder why Galileo brought it up at all, only to miss the point.

Too much philosophizing can be dangerous, but this sort of philosophy – extracting results from speculative physical laws – is exactly what theoretical physicists do. The name of the game for a theoretical cosmologist, for example, is to come up with some crazy ideas about how the universe might work, the way Sagredo came up with an idea about falling bodies. Then, the cosmologist tries to work out the consequences of the theory, for example that cannonballs ought to hang in midair until a slight breeze comes along and gives them a downward tap, and they come plummeting back to Earth extremely quickly. If the theory doesn’t agree with observation, it’s wrong.

One difference between the Renaissance and Internet Age versions is that Sagredo’s and Salviati’s theories about falling are easy to test. The experiment doesn’t require any equipment. You just drop something. If you don’t have a thing, you can try jumping instead. But with advanced theoretical ideas, it can be very difficult to make the required observations. That’s why we need giant particle accelerators and kilometer-long interferometers and thirty-meter telescopes and ridiculously-good gyroscopes.

But another problem is that working out the consequences of modern theories is hard. We saw an example of Galileo failing to work out the consequences of a theory, but that was a simple mistake, and if someone had brought his attention to it, he’d have been able to fix it. Some of today’s new ideas about physics are so complicated that even if we can state the law (the equivalent of Sagredo’s idea about velocity being proportional to distance fallen), we may not know how to get to the conclusion (cannonballs hanging over our heads).

We’ve come long way since Galileo, figuring out lots of ways to check ourselves and test our ideas. But there’s a much longer way left to go.