FOM: A problem in Foundations of Statistics
JoeShipman@aol.com
JoeShipman at aol.com
Thu Apr 18 23:43:55 EDT 2002
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
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