FOM: A problem in Foundations of Statistics

[JoeShipman@aol.com]

Quoth Todd Wilson:

>So what's wrong with the traditional solution involving so-called
>Bernoulli trials?  Without any other information, we assume that the
>event in question occurs with probability a/m; therefore, the chance
>that it will occur a further c out of n times is

>    C(n,c) * (a/m)^c * (b/m)^d ,

>where C(n,c) is the binomial coefficient "n choose c".

Todd, what's wrong with this is that it fails to take into account that other frequencies than a/m can lead to a successes in m trials.  For example, let a=m=1, n=2.  Then you are saying that having observed 1 success in one try, you are certain that you will observe 2 successes in the next 2 tries.  Similarly, if you'd observed 1 failure in 1 try, then you would be certain the next 2 tries would be failures.  Professional statisticians prefer not to draw such extreme conclusions.

An obvious desideratum is that as m approaches infinity, the chance attributed to the event "c successes in the next n tries" approaches the binomial formula you give.  But for m small other factors are also important.

I don't mean to completely dismiss your answer, it's one of several reasonable ones, but you did ask what was wrong with it.

-- Joe Shipman