Let $k$ be a known number of groups and let $X_1,\dots X_N$ be iid random variables each with density $f$ with regards to the counting measure.
$$ f(x) = \sum_{i=1}^{k} 1_{\{i\}}(x) p_i $$
($1_{\{i\}}(x)$ denotes the indicator on the $i$'th group)
This means that the joint likelihood for the $N$ observations will be
$$ L_{X_1,\dots,X_N}(p_1,\dots,p_k) = \prod_{j=1}^N \left(\sum_{i=1}^{k} 1_{\{i\}}(x_j) p_i\right) $$
Intuitively, I would think that the best guess for each $p_i$ would be the number of observations in group $i$ divided by $N$. Do we have any meaningful way of proving that this is a good estimate?