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Suppose that a factory produces coins whose bias θ follow a Beta distribution with parameters α=5 and β=10. If a random coin was chosen from the factory and tossed 100 times which resulted in 50 heads and 50 tails. What is the probability of coin being fair?

My approach:

Prior : Beta(5,10)

Posterior : Beta(50+5,50+10)

Now coin being fair is θ=0.5, But the probability of point in pdf is 0. I think the answer to 'probability of coin being fair' is 0. But still confused.

Am I missing something?

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I think you've made a small type in your answer, the posterior is Beta(55, 60).

That aside, I think the question is (perhaps intentionally) vague. It requires you to do more than just maths, you need to interpret as well. You could argue that a coin is fair if $0.5 - \epsilon < p < 0.5 + \epsilon$, and then calculate

$\int _{0.5 - \epsilon}^{0.5+\epsilon}\beta(q ; 55, 60)dq$

for a range of $\epsilon$.

Or you could take the hardliner approach and say the probability of the coin being perfectly fair will always be zero if you use a Beta prior (as far as I can tell, you need a prior which has some sort of delta function like properties, i.e. some finite mass concentrated at a point, for there to ever be a finite probability of the coin being exactly fair)

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  • $\begingroup$ Thanks. It was really helpful. PS I corrected the type $\endgroup$ – Shiva Shankar Aug 26 at 11:58

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