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Here's an even simpler arrival-interval puzzle, which doesn't
require a Geiger counter or dice or anything else.
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.
Question 1a: 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 1b: 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 start looking at a random time, how long
do you have to wait for the passage of the next bus, on average?
Question 2b: 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?
Final question: How do you reconcile the various answers? Never
mind the math; explain what's going on in conceptual, qualitative terms.