Re: Simulating radioactive decay.

[Robert A Cohen <bbq@ESU.EDU>]

On Fri, 14 May 1999, Ludwik Kowalski wrote:

> Robert A Cohen wrote:
>
> > ... After one throw, there are (1-p) left.  After n throws, there
> > are (1-p)^n left.  We want to know the number of throws, n,
> > that leave us with 1/2 left:  (1-p)^n = 1/2
> >
> > Solve for n to get n = - ln(2) / ln (1-p)
>
> Very good. Too bad I could not do this myself yesterday.
>
> > So, what is the discrepancy caused by using p rather
> > than ln(1-p)?
>
> That is not what I was referring to. I was talking about
> using lambda instead of p. For p=1/6, according to your
> formula, the half-life n= 3.801 throws. Use this to calculate
> lambda = ln(2)/n=0.245. That is the paradox. We started
> with p=1/6 (each pencil has six sides) and we concluded
> that the probability of decay per unit time is not 1/6.
>
> My point was that we should not identify p with lambda.
> By definition, lambda is the probability of decay per a
> negligibly small unit time. On the other hand, p, is the
> probability that a pencil will land with its label up. It is
> also a probability of decay per unit time but that unit is
> not small in comparison with n. Now I am using your
> notation for the half-life, n; yesterday I referred to it as T.

"p" is the same as "lambda".  A Taylor series expansion of ln(1-p)
gives...

   ln(1-p) = - p - p*p/2 - p*p*p/3 - p*p*p*p/4 - ...

Thus, replacing ln(1-p) with p is only valid when

        p/2 + p*p/3 + p*p*p/4 + ... << 1

Or, at least, p must be very small in comparison to 2

If p = 1/36, then the error shoud be about 1/72 or 0.0138 or 1.4%.

As Brian Whatcott pointed out...

> T = ln(2) / (1/36) = 24.95 rolls
> T = ln(2) / ln(1-1/36) = 24.61 rolls

The percent error is 100%x(24.95-24.61)/24.61 = 1.4%


Of course, one could then ask why we use
            T = ln(2) / p
instead of the more accurate
            T = -ln(2) / ln(1-p) ?

The answer, I believe, is that the expression comes from:

     dN/dt = - p N = - lambda N

which assumes continuous changes in N, not discrete changes.

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| Robert Cohen               Department of Physics       |
|                            East Stroudsburg University |
| bbq@esu.edu                East Stroudsburg, PA  18301 |
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