# Posterior beta distribution and probability of point

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?

• If you pick any hypothesis that $\theta$ is a real line, probability of it will be always zero by design. You need to pick something like ROPE cran.r-project.org/web/packages/bayestestR/vignettes/…
– Tim
Aug 26 '20 at 9:45
• Thanks. I will go through it. Aug 26 '20 at 11:57

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 $$\epsilon$$.