In Bayesian probability, does the prior distribution $\pi(\theta)$ only depend on $\theta$? For example, suppose the prior distribution of the unknown parameter $\theta$ is binomial. Then does $$ \pi(\theta) = \binom{n}{\theta} p^{\theta} (1-p)^{n-\theta}$$
Whereas if $f(\theta|x_1)$ is binomial then $$f(\theta|x_1) = \binom{n}{x_1} p^{x_1}(1-p)^{n-x_1}$$
Is this correct?