Posterior for a coin whose bias is uniform

The probability that a coin lands heads is itself random, drawn from a uniform distribution on $[0,1]$. (a) After one toss that comes up heads, what is the posterior distribution of the heads probability? (b) What is it after observing $750$ heads in $1000$ tosses?

(a) Beta$(2,1)$, density $2p$; (b) Beta$(751,251)$. The uniform prior is Beta$(1,1)$ and the binomial likelihood is conjugate, so each head adds one to the first parameter and each tail one to the second. In (b) the posterior mean is $751/1002\approx0.75$ with standard deviation about $0.014$, so it is close to normal around $0.75$.