Is the coin really fair?

A coin is selected according to one of two equally likely hypotheses: • $H_0$: The coin is fair, so the probability of heads is exactly $p=1/2$. • $H_1$: The probability of heads $p$ is initially unknown and follows a uniform distribution on $[0,1]$. The coin is tossed independently six times, conditional on its value of $p$, and exactly three heads and three tails are observed. Using Bayes' theorem, determine the posterior probability $\mathbb P(H_0\mid E)$, where $E$ denotes the observed event. If this probability is expressed as a fraction $\alpha/\beta$ where $\alpha$ and $\beta$ are coprime positive integers, what is the value of $\beta$?