arrival times; was: Geiger (a challenge)

[John Denker <jsd@MONMOUTH.COM>]

At 09:03 AM 7/8/00 -0400, Ludwik Kowalski wrote:

> The exponential distribution [of arrival intervals] may be counter-intuitive.

Here's an even simpler arrival-interval puzzle, which doesn't require a
Geiger counter or dice or anything else.

Part 1:  We have a bus stop.  Busses arrive every ten minutes like
clockwork, all day and all night.  If the last one was at 8:15 the next one
will be at 8:25, guaranteed.
  Question 1a: If you show up at the bus stop at a random time, how long
do you have to wait for the arrival of the next bus, on average?
  Question 1b: If you show up just as the previous bus is leaving, how
long do you have to wait for the arrival of the next bus, on average?

Part 2:  Same as above, except the bus arrival events are random.  In fact,
the events are IID (Independent and Identically Distributed over
time).  The average rate is one every ten minutes, i.e. an average of 6 per
hour or 144 per day or whatever.
  Question 2a: If you show up at the bus stop at a random time, how long
do you have to wait for the arrival of the next bus, on average?
  Question 2b: If you show up just as the previous bus is leaving, how
long do you have to wait for the arrival of the next bus, on average?

HINT:  The answers to part 1 are not identical to the answers to part 2.

Final question:  How do you reconcile the various answers?  Never mind the
math;  explain what's going on in conceptual, qualitative terms.