Probability Philosophy and Applying Inference

Date: 10/11/2004 at 12:44:50
From: John
Subject: probability and statistical inference

If I flip a coin 4 times and they all turn out to be heads, what is 
the probability that the coin is fair? 

Because I am not sure if there is a proper comparison distribution, 
would I have to use the T-distribution and is this problem even
answerable?  Would the question be answerable if I flipped the coin 50
times and it turned out heads each time?  Any help would be great. 

Date: 10/11/2004 at 13:03:09
From: Doctor Schwa
Subject: Re: probability and statistical inference

Hi John,

If you're a "frequentist" probability philosopher, the question has no
answer: either the coin is fair, or it isn't--what's the repeated
event from which we can abstract a probability?  That is, you can say
"the probability of this die showing 6 is about 1/6" based on rolling
it a lot of times.

But you can't do the same for "this coin is fair" because it is either
always fair, or not--there's no variation!

On the other hand, a "Bayesian" probability philosopher would be
perfectly happy with your question.

They would ask for one more piece of information first, though:  what 
do you know about the person giving you the coin?  How much did you 
trust them?

The Bayesian would then use P(coin is fair) in the abstract, followed
by P(4 heads in a row | coin is fair) compared with 
P(4 heads in a row | coin is unfair), to eventually determine
P(coin is fair | 4 heads in a row).

More likely than any of the above nonsense, though, what your teacher
really wants you to calculate is a P-value.

A P-value is NOT the probability that the coin is fair, though many 
people (and many statistics textbooks, even!) often misstate it as such.

A P-value is just one of the probabilities that a Bayesian would use:
P(4 heads in a row | coin is fair).

That is, you can answer the question "how likely is a fair coin to
produce 4 heads in a row", and if that P-value is small enough,
decide to reject the hypothesis that the coin is fair--but it's
certainly not the probability that the coin is fair!  You're simply
casting doubt on the ASSUMPTION that it's a fair coin by noticing that
the DATA you got would be really unlikely if the coin was fair.  It's
the DATA (the 4 heads in a row) that have a really low probability.

I hope that response helps!

Enjoy,

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/