I am struggling with a voting model problem that is set up as follows:
Suppose there is a binary issue where policy $L$ and policy $R$ are equally likely to be optimal. There are three voters who vote for the optimal policy with probability $p = 0.75$. What is the probability that the optimal policy receives 3/2/1/0 votes? If policy $R$ is chosen unanimously, what is the conditional probability that it is the optimal policy? What if $R$ wins with 2 votes?
It seems like the initial voting probabilities should be calculated normally. For example, the probability that the optimal policy wins with three votes is $ 0.75*0.75*0.75 = .421875$. However, if I compute the probability in that manner then the probabilities for each outcome (3/2/1/0 votes) do not sum to 1, and it seems like they should. I'm not even sure how to approach the conditional probability for the second part of the question.