I've been reviewing probability and came across this problem:(adsbygoogle = window.adsbygoogle || []).push({});

A penny may be fair or it may have two heads. It's tossed n times and it comes up heads on each occaision. If our initial judgement was that both options for the coin ("fair" or "both sides heads") were equally probable, what is our revised judgement in light of the data?

The book's answer is P(fair)[itex]= \frac{1}{1+2^n}[/itex], but I don't understand this. Certainly, the book's answer makes sense for n=0 (before we toss the coin).

Suppose we toss the coin once and it comes up heads. The probability for this outcome is 1/2. According to the book's answer, P(fair)=1/3.

I guess what I'm having trouble with is connecting the probability for an event to happen (say, tossing n heads) with the probability that the coin is fair.

What is the connection between these two probabilities? How is the book's answer justified?

Thanks!

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# Determining the fairness of a penny

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