Assume the following situation: there are N people voting for or against M candidates (N >> M > 1). Every person $i \in \{1, ..., N\}$ has to vote with yes or no for every candidate $j \in \{1, ..., M\}$. The probability of voting "yes" for a given candidate $j$, is is the same for all N people. The winner of the election is the candidate with the most "yes" votes.
The voting procedure is as follows: one by one person is called to publicly announce his yes/no votes for all M candidates.
At some given point in time, n out of N people have announced their votes. Therefore we know which candidate has how many votes at that moment.
Task: find the probabilities $p_j$ of every candidate to win the election, given the current election results.
I assume that there exists some modification of Bayes' formula for calculating this, right?
My attempt: after n people have announced their votes, all candidates will have some fraction of "yes" votes between 0 and 1 (let us call them $\mu_j^{n}$). We can approximate the final fraction of "yes" votes as a normal distribution with mean=$\mu_j^n$ and some std=$\sigma_j^n$:
$\mu_j = \mathcal{N}(\mu_j^n,\sigma_j^n)$
The problem is, I don't know how to calculate $\sigma_j^n$.