I know a number of people, say 100.

Every year I visit 3 cities (Vienna, Paris and Cairo) once. In each of
those cities there are say 250 places I could be in, and on any one
visit I will probably visit 125 of them (eg the train station, 5
cafes, 10 shops etc, parts of a street). Let us define a place as a
circle of radius 10 metres.

Likewise each of my 100 known people. They visit those three cities
(their visits have no relation to mine, or to each others). We may or
may not be in those cities at the same time, and if we are we may or
may not be in the same place within that city at any time. Our trips
are totally independent and we do not know each others travel plans.

What are the chances of me bumping into any one of my known people -
ie that I and any one of those 100 will, within the same 60 seconds,
be in one of those 125 places of one of those 3 cities.

(I would like to see the calculation, not just the number).

Thank you.

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Request for Question Clarification by smudgy-ga on 25 Jun 2003 21:39 PDT

Interesting question.

As it stands, without a little more knowledge about some of the
details of the problem, it would be nearly impossible to answer.
Either some more detail about the duration/potential timing of the
visits, or a simplification in that regard, would be necessary to
effectively answer the problem. For instance, if each trip were ten
days long and could happen at any time during the year, that would at
least be something to work with. A far easier simplification would be
to say that you and your friends tend to have the same vacation
periods, so that during three well-defined periods, you and your
friends are all on vacation simultaneously. But some kind of
clarification is necessary in this regard before the question could
adequately be answered, I believe.

Thanks,
smudgy.

---

Clarification of Question by macaonghus-ga on 27 Jun 2003 13:02 PDT

Each visit to a city may last say 1-7 days BUT that isnt so relevant -
what matters is the visit to the common places. Each visit to a place
(ie a circle of 10 metres, I think I said) lasts from 1 minute to
three hours, depending on whether I am walking through it or it is a
restaurant, for example. The visits may happen at any time of year -
all the people involved visit at random times.

Any questions, let me know.

I can't help you with the calculation, but I CAN tell you that it
happened to ME, not once, but TWICE - in the same day - within one
minute's time!

I attended rural high school in the US, graduating with a class of
only 24 students. Three years later I found myself living and working
in Germany. One day while shopping in a grocery store in a town I was
touring about 60 miles from town I lived in, I rounded the corner and
literally ran into a man I went to high school with. He was also
living and working in Germany (unbeknownst to me) and just happened to
be touring the same town, on that same day, more than 100 miles from
where he lived in Germany. We shook hands and marveled at the
coincidence, then I quickly turned to fetch my wife and literally
bumped into a girl with whom I had attended the same high school. I
was dumbfounded - then my friend told me that he had married this girl
before coming to Germany, and of course, she came with him.

What are the chances of that? Weird huh?

Good luck;
tutuzdad-ga

No help with calculation here either, but it also happened to me
twice.  Not on the same day but on two separate trips to Israel. 
Strange but true!

tlspiegel

It's never happened to me.

Therefore, the odds are 2 to 1.

LOL Probono

Actually, calculating the chances for this only proves that statistics
is not that accurate science. If we calculated the chances that 1/100X
would be in one of 3Y and would be in one of 750 places (or 250 per
Y); and the possibilities of such an event, we sould get a certain
numbe. However, it seems that in real-life (TM) there's more
probablity you'll meet people of your own milleu, because they are
more likely to travel to the same places as you.

I'll be a wise acre.

50% -- It either happens or it doesn't.

:-)

Well, I think it's time for someone to actually have a bash at this...
just don't attach too much meaning to the answer!

First of all, the length of trips is very important. This directly
impacts the chances of you and your friends being in the same city at
the same time, which is an important precondition for meeting. ;-)
Thus, if you are in city X on a particular day, and one friend visits
that city for an n-day stay in the given year, they have an n/365
chance of being in city X on that day. Since you've said 1-7 days,
I'll assume that stays of 1, 2, 3, ..., 7 days are equally likely,
both for you and your friends. Thus the median trip would be 4 days.

Now, you've said that on a typical trip you might visit 125 (of 250
possible) places. It clearly doesn't make sense to assume that this is
independent of the length of the trip, so I'll assume a constant rate
of visitations of 30/day. Again, this applies both to you and your
friends.

Now, the next part is one of the most crucial. We need to describe the
distribution of time spent in each place. You've said that this can be
from 1 minute to 3 hours, but obviously they can't be uniformly
distributed; at 30 visits per day, with probably not much more than 15
hours a day allocated to them, we need to have a mean duration
preferably no more than 30 minutes.

Perhaps not surprisingly, I haven't been able to come up with a
terribly convincing piecewise smooth function to represent this (I've
only played around with functions in two pieces; I suspect three are
needed). So here's a sample distribution I made up out of thin air:

t(min)   Ave # per day
------   -------------
  1        5
  3        5
  5        5
 10        4
 15        3
 30        2
 45        1.5
 60        1.25
 90        1
120        1
150        0.75
180        0.5

This gives an average duration of just under 25 minutes, or around
12.5 hours' worth of visits per day.

One difficulty with this is that the duration of your stay in each
place is likely to be well correlated with the duration of your
friends' visits to the same place. If you spend five minutes at the
train station and three hours in a restaurant, it's much more likely
that your friends will do the same than the reverse. (I suppose their
train could break down, and they might decide not to eat at the
restaurant (especially after seeing the prices?) - but it's surely not
as likely.) So I've drawn up a possible joint distribution of times
and put it on the web at
http://home.iprimus.com.au/scarletmanuka/friends.htm for your
edification (or whatever) - it was a bit too awkward to insert here.
Note that the table entries should be divided by 30 to give a
probability distribution. The significance of this table is that it
represents how long each of you are visiting a place that you both
visit. Of course, the numbers here are also just made up.

Now we're beginning to be in a position to start answering the
question!

The time of day that you and your friends visit certain places will
also be somewhat correlated in real life, but I am going to ignore
this, except to confine visiting activities to a 15 hour period
instead of around the clock.

So, assuming that you and a given friend are both in the same city on
the same day and decide to visit the same place, what are the chances
that you'll be there at the same time? Well, if you visit for x
minutes and your friend for y minutes, there are xy possible
combinations of starting times (more or less - this is where a
continuous distribution would simplify things) that will lead to a
meeting, out of (15*60)^2 = 810,000 possible combinations, assuming
1-minute granularity to time. ;-)

Now if we divide the table at the link I gave earlier by 30 to get a
probability distribution, and multiply each element by the respective
value of xy/810000, the sum of all the table entries gives us the
probability that you meet your friend, given that you both visit the
same place on the same day. Excel tells me that for these figures this
probability is 0.00258 (of course, given our assumptions, there's a
considerable fudge factor here).

So, if you and your friend are in the same city on the same day,
what's the expected number of places that you'll both visit that day?
You're both visiting 30 places out of 250 - they won't actually be a
uniformly random selection, but I'll pretend they are. The probability
of you both visiting a fixed place is then (30/250)^2 = 9/625, so the
expected number of places you both visit is (9/625)*250 = 3.6; so the
expected number of meetings is 3.6*0.00258 = 0.0093.

OK, so what is the expected number of days that you will both be in a
given city? Let's assume your visit to the city is N days long and
your friend visits for M days. Then the expected number of overlaps is
MN/365; since M varies uniformly from 1 to 7 you can expect to meet
your friend (4*N/365)*0.0093 = 0.00010*N times during your stay.

Now we've got the average behaviour for any given friend, so we
multiply by 100 to say that your chance of meeting any friend during
your visit is 0.010*N.

From there it's simple: the average visit is 4 days and you make three
visits per year, so you can expect around 0.12 meetings per year, or
roughly one every eight years.

So there you go! Feel better now?

--
The Scarlet Manuka

Actually, there are more confounding factors than Manuka has supposed.

How good are their eyesights? How good their memories?

What are their sexes? (Women change their appearance 10 times a day,
so that even their next-door neigbours fail to recognise them.)

How gorgeous are they?

In fact, the more I think about it ... The more I realise that it's
just impossible.

Hey, he only asked for the chances of meeting them, not for the
chances of actually recognising them when he does! ;-)

Of course this does add some difficulty to the task of verifying my
oh-so-rigorous derivation... <grin>

Thank you all, especially obviously manuka....

I just remembered that, a few years ago, I attended a party in The
Hague ...

During the course of the evening, I spoke separately to two Dutch
women each of whom told me that she had used to live in Surinam.

Later, I introduced them to each other and (would you believe?) they
had even known each other when they had lived in Surinam ... but they
had failed to recognise each other that evening!

That must prove something ....

Perhaps that all statisticians are mad?