Hello, this problem is somewhat based on the game Among Us.

agrimm2

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Hello, this problem is somewhat based on the game Among Us.

There is a space station with 3 human crewmates and 1 alien impostor. Humans do not know who the impostor is. But the impostor knows the identity of each person. Each day, the remaining people come together and vote on who the impostor is. The person with most votes gets kicked out. Nobody gets kicked if there is a tie among those with the maximum number of votes. The impostor wins if he/she is still in the station and there is only 1 crewmate left.

What is the probability of the impostor winning if every day, everyone, including the impostor, votes to kick out someone other than themselves at random with equal probabilities?

I saw this problem on another site and was curious about how to solve it.
 
After two votes that see a crew member kicked off we are left with two winners remaining on board. By symmetry the alien has a 50 / 50 chance of being one of them.
When analysing the problem we can find in the first round 81 distinct ways in which votes can be cast but 21 of these produce a tie so just lead to another vote. Similarly we ignore two of the eight distinct ways in the second round.
 
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