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
-- 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
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
*) This is sometimes called "Kolmogorov" algorithmic
complexity, but that is very unfair to Chaitin.