Suppose we live in a country with an interesting electoral system: each electorate has exactly 3 voters. 2 parties run for office, and each voter has a 50/50 chance of voting for each party. Whoever wins the majority of electorates wins the election overall. Given an arbitrarily large number of electorates, what is the probability that the party that won the election lost the popular vote? (This is a more specific version of my earlier question Probability of winning an election while losing the popular vote)
My 'brute force' computer model yields an answer of very close to 1/6th. Does anyone have ideas for how to solve this problem analytically?