John D. wrote in a note dated June 24th
(and which I seem to have missed on my home mailer)
>...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.
Or yes, as the case may be.
[though that was not the question posed, at least in
my rendition of it, given again below...]
>> The question: what is the probability that the car passed the
>> same spot at the same time of day, going both ways?
It is not necessary to suppose that a particular route
has divided lanes. Besides the obvious one-lane route of
which a great many exist in the US as dirt roads, there
are still examples of the three lane highway here with an
overtaking center lane (danger!)
But finding pathological reservations to the
"passing the spot" clause misses the humor, I fancy.
> ....most physicists
> have a treeemendously good intuition about probability,
> based on notions of coin tossing et cetera.
Hmmmm.....if one can visualize two cars passing the same point
at the same time on the same day in opposite directions,
just how much intuition does it take to decide how certain
it is that they have passed this point at that particular
time on those consecutive days? :-)
> -- 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.
This is in effect, an amusing restatement of the paradox given in
Aristotle's Physics and attributed to Parmenides or
Zeno of Elea - about Achilles and the Tortoise - concerning
the impossibility(?) of Achilles passing the tortoise when they race.