Re: [Phys-l] Poisson stat.

[Hugh Haskell <hhaskell@mindspring.com>]

At 08:09 -0700 04/17/2009, John Denker wrote:
> The per-die process is a Bernoulli process (subject to the
> usual mild assumptions, e.g. that the dice are not damaged
> by the throws).
>
> The overall experiment is a good model of radioactive decay.
> Quite a direct model.
I agree that throwing dice to simulate radioactivity is a 
pedagogically valuable tool. It is a good model of radioactivity. 
Good, but not excellent.

Limiting ourselves to dice as the "radioactive" substance, and using 
the usual formulas for half-life, we know that half-life (T) can be 
expressed as

	T = (ln 2)/p		(1)

where p is the probability of radioactive decay per unit time, and T 
will then be expressed in the same unit of time as p. If p is 1/6 (as 
in the dice simulation), the we find T = 4.16 units of time (normally 
called "throws" when doing the dice simulation).

If we do the simulation in a real classroom with several groups of 
students each doing several repetitions of the "experiment" (lots of 
dice required for this), and we combine all the data to get good 
statistics of the "half-life," it will turn out that the results will 
converge to about 3.8 throws, a difference from the expected value of 
4.16 throws of a little over 9%. If the number of trials is large 
enough (one class of several groups will suffice) this difference 
will be easily observed.

Hopefully, a bright student will ask why the result is that far from 
the expected value.

Repeating the experiment will lead to similar results--the measured 
half-life always about 10% less than the expected value. The vagaries 
of probability are not to blame, suggesting that there must be a 
systematic error in the experimental design.

The reason for the difference comes from the fact that time can be 
divided into essentially infinite segments while "throws" are by 
their nature integers that cannot be changed. The probability of half 
of a (large) set of dice coming up any given number in m throws is 
given by

	P = (1 - p)^m		(2)

where P is (in this case 1/2) the probability getting the desired 
fraction of a given number in m throws and p is the probability of 
any given number turning up on any given throw, in this case 1/6.

Solving equation (2) for m (i.e., the number of throws) gives

	m = - (ln 2)/[ln(1 - p)]	(3)

For p = 1/6, equation (3) gives m = 3.8 throws.

And just to close the loop, expanding ln(1 - p) in a Taylor's series 
will show that equation (1) is an approximation of equation (3) for 
small values of p. In other words, simulating radioactivity by 
throwing a set of D&D dice with 20 sides will give an answer much 
closer to that found when a radioactive isotope has a decay 
probability of .05 per unit time (about 2.5%).

So maybe we should start collecting sets of those 20-sides dice and 
use them for the radioactivity simulation.

Hugh

PS: I submitted a short paper based on this analysis to The Physics 
Teacher, but, in their infinite wisdom, they declined to publish it.

--
Hugh Haskell
mailto:hugh@ieer.org
mailto:hhaskell@mindspring,.com

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