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/
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