Probability of $\max_i \{X_i\} = X_0$ where $X_i$ are iid binomial We have $M$ Binomial random variables, where $X_0 \sim $ Bin$(n,p)$ and $X_i \sim $ Bin$(n,1/2)$. 
Suppose $p > 1/2$. I'm interested in the probability that $\mathbb{P}(\max \{X_1,\dots,X_M\} \geq X_0)$. Is this tractable? 
If not, is it tightly boundable/approximable? If this is a very difficult question, I'd accept a reference providing some insight on the problem as well.

The maximum of $\{X_i\}_{i\in\{1;M\}}$ will be at least as great as $X_0$ when it is not that all values are less than $X_0$.   (Employ the rule of complements.)   Assuming that $X_0$ is also independent of the iid variables then:
$$\begin{align}
 \mathsf P\left(\max_{i=1}^M\{X_i\} \geqslant X_0\right) =&~ 1-\sum_{k=1}^n\mathsf P(X_0=k)\mathsf P\Big(\bigcap_{i=1}^M X_i<k\Big)
\\ =&~ 1- 2^{-nM} \sum_{k=1}^n\binom{n}{k}p^k(1-p)^{n-k}\left(\sum_{h=0}^{k-1}\binom{n}{h}\right)^M
\end{align}$$
