You are looking into the quality of the diamonds of the diamond mines in a new area. You have a special interest in "Fancy diamonds" 5 of quality IF and VVS, and you are estimating \(\pi\), the proportion of diamonds from the new mines that fit one of these descriptions. After having evaluated 172 diamonds, you have found 19 that are either IF or VVS, while the rest are of lower quality grades.

(a) You are unsure which of the 3 neutral priors to choose. Find the posterior distributions you find for \(\pi\), for each of them.

(b) Calculate, for each of the 3 priors, the probability that \(\pi>0.1\).