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Re: clarification: arrival times; was: Geiger (a challenge)

John Denker wrote:

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?

5 minutes

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?

10 minutes (always 10 minutes)

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?

This is the exponential distribution of arrivals. The mean time
is 10 min. Thus the probability of arrival per unit time (per minute)
is p= 1/10. The probability of waiting t minutes is given by

W(t)=Const*exp(-p*t).

The integral of W(t), from zero to infinity, must be unity. For that
reason W(t)=0.1*exp(-t/10).

For example, W is 0.1 when t=0, it is 0.061 when t=5 min and
0.008 when t=25 min, etc. How long do I have to wait, on the
average? Ten minutes.

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?

Same answer as in 2a.

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

One bus knows nothing about other busses, or about me.

NOW THE OTHER ISSUE. SOMEBODY COUNTS
RANDOM BUSSES PASSING IN EVERY 1/2 HOUR.
ON THE AVERAGE 3 BUSSES ARE EXPECTED.
5000 HOURS PASSED, WHAT IS THE EXPECTED
DISTRIBUTION BUSES PER THIS TIME INTERVAL?

HOW MANY TIMES ZERO BUSES ?
HOW MANY TIMES ONE BUS ?
HOW MANY TIMES TWO BUSES ?
HOW MANY TIMES THREE BUSES ?
HOW MANY TIMES FOUR, FIVE, SIX ETC.

MY POINT WAS THAT GEIGER COUNTS ARE HIGHLY
RELIABLE SOURCES OF CLEAN DATA. TRY TO COUNT
REAL BUSSES AND ALL SORT OF UNUSUAL OUTCOMES
WILL BE OBSERVED, FROM TIME TO TIME.

THE QUESTION WAS ASKED WHY DO WE NEED
EXPERIMENTS WHEN IDEALIZED SIMULATIONS CAN
BE PERFORMED WITH EXCEL? THIS IS PHILOSOPHY
OF PHYSICS, NOT PHYSICS.