Given a specific probability $p$ of heads, the probability of getting $h$ heads and $t$ tails is just the binomial distribution: $P(H = h, T = t | n, p) = p^h (1-p)^t {n \choose h}$ (though we consider $p$ as varying, rather than $n$ and $h$.
With a uniform prior $g(p) = 1$, that probability is just the weight.
The integral of this is $\frac{1}{1 + h + t}$, giving a probability density
of $(1 + h + t) p^h (1-p)^t {n \choose h}$. The mean is the integral of $p$ times this, is $(1 + h + t) \int p^{(h+1)} (1-p)^t {n \choose h} dp$. Reusing the result from last time, we know that this must be $(1+h+t) {n \choose h}/{n+1 \choose h + 1}/(2+h+t) = (1+n)(h+1)/(n+1)(n+2) = (h+1)/(n+2)$. This is slightly "hedged toward the center" from the naïve estimator $h/n$ (which is the peak of the distribution).
Another common prior that a Bayesian might use is the beta distribution. It's handy because it is a conjugate prior for the binomial distribution. After collecting data generated by the binomial distribution, the probability is still in the form of a beta distribution. In fact, the uniform prior is just the beta distribution with $\alpha = \beta = 1$. Heads and tail each just add one to the parameters $\alpha$ and $\beta$ respectively. The integrals were essentially worked out above -- factorials generalize to $\Gamma(x+1)$. It's often considered that this case of $\alpha = \beta = 1$ is too conservative, and that $\alpha = \beta = 1/2$ "assumes less" and "lets the data speak more".
With the Uniform Prior (Beta(1,1)), $\overline{p} = (h+1)/(n+2)$:
$C_1$, 3 heads, 0 tails: $\overline{p} = 4/5$
$C_2$, 7 heads, 3 tails: $\overline{p} = 8/12 = 2/3$
$C_1$ is expected to do better.
With the Beta(1/2, 1/2) prior, $\overline{p} = (h+1/2)/(n+1) = (2h+1)/(2n+2)$:
$C_1$, 3 heads, 0 tails: $\overline{p} = 7/8$
$C_2$, 7 heads, 3 tails: $\overline{p} = 15/22$
$C_1$ is expected to do better.
Actually calculating the chances of $C_1$ being better than $C_2$ involve a rather nasty integral, but the calculated $p$ value is enough to tell you which is the better bet.