How to calculate the chance of a coin being unfair?

Question

Say I have $x$ heads out of $n$ samples/throws. Say $x>\frac n2$.

Based on the sample size $n$ and the fraction $\frac xn$ what is the probability that the coin is unfair AND unfair towards heads?

Sorry if this is a stupid question, it's just that most binomial examples seem to deal with the probability of getting say heads, but I'm interested in the confidence level that the coin is unfair towards heads.

Answer 1

"Chance of a coin being unfair" is too vague and needs to be more specific.

• The p-value of a binomial exact test is the probability of seeing more than $x$ heads if the coin were fair. [Alternatively you can use a normal approximation to approximate this probability (z-test).] Although this test could in principle help you detect whether the coin is unfair towards heads, it isn't really "the probability that the coin is unfair."
• In the frequentist setting (like the above), the fairness of a coin is fixed (but unknown); it either is fair or it isn't (so it does not make sense to ask the probability of it being unfair). On the other hand, the Bayesian setting [that Matthew Pilling mentions] treats the coin's probability of heads as a random quantity [following some prior distribution on $[0,1]$], so then it makes sense to talk about the probability of heads. But given that you brought up the binomial test, I am guessing this isn't what you meant to convey.