Suppose I have a bag of $n$ biased coins, and I sample without replacement $m < n$ of them, and measure each coin $i$ for their parameter $p_i \in [0,1]$, that is each coin is $Bernoulli(p_i)$. Now I would like to ask what is the most likely $p_{m+1}$ on the next coin I pick. I am not sure how to answer this question aside from taking the mean of all $m$ coins' parameter so far. That is: $\hat{p}_{m+1} = \frac{p_1 + \ldots + p_m}{m}$.
However I cannot prove why this is the most likely $p$. Any thoughts?