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*From*: John Denker <jsd@av8n.com>*Date*: Tue, 24 Jun 2008 16:25:27 -0700

There are additional layers to this puzzle that haven't been

discussed yet. The problem is that the question is either open

to interpretation, or at best highly sensitive to the details

of the wording.

I don't know the canonical wording. I searched the cartalk

web site and didn't find it. But that almost doesn't matter,

because there is known to be a family of seemingly-similar

questions, and we can fruitfully discuss the family.

1) One question of interpretation is exposed by the tactic of

considering two travelers on the _same_ day. In the normal

course of things, they do not in fact collide, because one

is in the northbound lane and the other is in the southbound

lane. They miss by many meters. So if the question asks

whether they are at _the same point_ at the same time, the

answer is no.

Maybe you think this defeats the spirit of the question,

but hey, if you're going to ask tricky questions, you have

a special burden to get the details right.

You could try to salvage the question by saying they pass

"abeam" the same point at the same time. But even then the

meaning is still sensitive to details. In particular, if you

ask whether they pass abeam a certain landmark at the same

time, that's a wildly different question, because not every

point is a landmark.

2) Another question of interpretation arises when the question

asks for "the probability" of this-or-that occurring. This is

a deep and fascinating line of discussion, because most physicists

have a treeemendously good intuition about probability, based on

notions of coin tossing et cetera. Once upon a time I had a

postdoc who had a brand-new Ivy League PhD in mathematics. At

one point I asked him to explain to me what a probability measure

was. He literally turned red and spluttered in frustration,

because he had had several experiences where I could in minutes

guess the answer to something that it would take him days to

calculate. The thought I was doing the calculation quickly.

He was astonished to learn that I couldn't do the calculation

at all ... which is why I wanted him to teach me. The funny

thing is, the physicist-style intuition about probability is

limited -- powerful but limited. By way of analogy, for 2000

years "everybody" thought Euclidean geometry was "the" only

geometry. Even after mathematicians discovered non-Euclidean

geometry, it was considered a mathematical curiosity, and it

was years later before physicists realized that the universe

is not in fact Euclidean. You cannot discover the geometry

of the universe by looking at axioms (Euclidean or otherwise);

the actual geometry must be measured.

So it is with probability. For centuries there has been a

school of thought that starts with the notion that "the"

probability distribution exists, and we just need to find

it. Sometimes one hears people talk about "the state of

nature". Alas that's not right. There are *lots* of different

probability measures, just as there are lots of different

non-Euclidean geometries. Just because you have one probability

measure in mind does not entitle you to assume that everybody

else has the same probability measure in mind.

Returning to the travel problem, there are lots of ways of asking

the question that invite people to analyze the problem point by

point.

-- Do the trajectories cross at point 1? No.

-- Do the trajectories cross at point 2? No.

-- Do the trajectories cross at point 3? No.

-- Do the trajectories cross at point 4? No.

-- Do the trajectories cross at point 5? No.

If you itemize all the points, and weight them equally, you conclude

that the probability of any given point being a "coincidence" point

is zero. Even if you correct for the size of the cars using some

sort of van der Waals excluded volume, the answer is still nearly

zero.

You may think this is an unsophisticated, foolish, or even smart-

alecky way to analyze the problem ... but I'm here to tell you it

is not wrong. Not at all. Basically this is Lebesgue measure,

and there are lots of exceedingly practical experimental and/or

theoretical situations where it is exactly the right approach.

The take-home lesson for this part (2) is that any time you hear

somebody talk about "the" probability, your defenses should go up.

You have to obtain additional information before you can answer

the question. Often you must ask things like "What is your

measure?" or "What is your ensemble?" If you're talking about

the ensemble of trips, the probability of a coincidence might

be 100%, whereas if you're talking about the ensemble of points,

the probability of a coincidence might be 0%.

Again I emphasize that this is not a trivial issue. People get

confused by it all the time ... and I'm talking about serious

professional people, not just first-semester physics students.

Here's a famous example:

2a) "What's the percentage of double stars?"

That is open to interpretation. Do you assign two units of measure

to a double star system, i.e. counting by stars? Or do you assign

only one, i.e. counting by systems? It's not like one answer is

right and the other is wrong; there are *lots* of different

probability measures, and you get to decide which one(s) you are

going to use. The tried-and-true rule for minimizing confusion

in situations like this is:

"Say what you mean, and mean what you say".

Also, remember you are free to choose, but keep in mind that

others may choose differently.

2b) Another example is Chaitin's * algorithmic approach to

probability. There is no such thing as "the" probability in

this approach. By dint of a highly technical definition it

is a "universal" probability distribution, but that does

not mean it is an all-purpose or general-purpose probability

distribution, or that it is unique or even approximately

unique. There are lots of different probabilities. I've

seen famous statisticians get this wrong, time and time

again.

*) This is sometimes called "Kolmogorov" algorithmic

complexity, but that is very unfair to Chaitin.

**References**:**Re: [Phys-l] C & C Trajectories***From:*Stuart Leinoff <leinoffs@sunyacc.edu>

**Re: [Phys-l] C & C Trajectories***From:*chuck britton <britton@ncssm.edu>

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