# Probability of predicting election winner while vote counting

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 $$\sigma_j^n$$.

• Caution: I am a beginner... Couldn't you model this with a Beta distribution? So the probability of winning for each candidate starts off as $1/M$ or as a Beta(1,M-1) but then after $n$ votes have gone by your new updated distribution for a given candidate to win is Beta with parameters Beta(1+k,M-1+n-k). Are you using the normal dist in your attempt due to CLT? Mar 26, 2019 at 19:39
• @HJ_beginner Thank you for your feedback. The reason why I am using Gaussian distributions is because of CLT and because this problem is part of a much more complicated problem in Machine Learning, that requires Gaussian variables :) Mar 26, 2019 at 19:44
• Thanks for the info, that's very interesting. I don't know anything about ML though I am very interested. Is it like a class you're taking or some algorithm? I hope someone else answers your original question. Mar 26, 2019 at 23:59