Suppose $Y_1, \dots, Y_n$ are i.i.d. bernoulli random variables. Also, $Y=\sum Y_i \sim {\rm binom}(n, \theta)$ and we have a prior beta distribution $\theta\sim {\rm beta}(a,b)$. I want to compute $P(\theta>0.3|\sum Y_i =k)$. Here is my work:
$P(\theta>0.3|\sum Y_i =k)=\dfrac{P(\sum Y_i =k|\theta>0.3)P(\theta>0.3)}{\int_{0.3}^{1} P(\sum Y_i =k|\theta>0.3)P(\theta>0.3)\mathrm d\theta}$.
My computation is according to Baye's rule but does not seem correct, especially the denominator. The "$\theta>0.3$" distorts everything (usually we have $\theta$ equals something instead). I am also wondering if there is a way to get rid of $\theta>0.3$ in $P(\sum Y_i =k|\theta>0.3)$, or at least write it in another way.
Thanks in advance!