Here's a version that makes the point even more clearly makes the
point. See the final paragraph below.
Part 1: Suppose busses drive past your window 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. You have a colleague who sits by the window
and keeps track of the busses.
Question 1a: If you walk up to the window at a random time and ask
your colleague how long it's been since the last bus, what's the answer, on
the average?
Question 1b: If you start looking at a random time, how long
do you have to wait for the passage of the next bus, on average?
Question 1c: If you start looking just as one bus is passing, how
long do you have to wait for the passage of the next bus, on average?
Part 2: Same as above, except the bus passage 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 walk up to the window at a random time and ask
your colleague how long it's been since the last bus, what's the answer, on
the average?
Question 2b: If you start looking at a random time, how long
do you have to wait for the passage of the next bus, on average?
Question 2c: If you start looking just as one bus is passing, how
long do you have to wait for the passage of the next bus, on average?
Standard hint: This is not a word game. This is not a trick
question. Everything I'm saying here is to the best of my knowledge true
and non-misleading. There's real science here, and you will see the answer
if and only if you understand the science.
HINT: The answers to part 1 are not identical to the answers to part 2.
How do you reconcile the various answers? Never mind the math; explain
what's going on in conceptual, qualitative terms.
In particular, show that the answer to question 2a is 10 minutes, and the
answer to question 2b is also 10 minutes. If these two bus-passages are 20
minutes apart on average, how do you reconcile that with the fact that
there are (on average) 6 busses per hour?