Probabilistic Voting with 3 Ballots

You want to vote for party A with probability p and party B with probability q = 1 - p. You are alone in the voting booth, where there are paper ballots marked A and B, exactly one of which you must put in the given voting envelope.

You want to do this with zero knowledge about your actual vote, i.e., such that you will only know that you have voted with probabilites p, q and nothing else (formally, the posterior probabilities must always be p, q). Finally, you are not allowed to deform the ballots or mark them in any way (as it will disqualify the vote).

For p = 2/3, you can easily do it with two A-ballots and one B-ballot: turn them upside down, mix them, choose one at random (i.e., uniformly) and put it in the envelope, and finally dispose of the other two (in such a way that you won't see what they are).

QUESTION 1. Can one do it for p = 3/4 with two A-ballots and one B-ballot only ?

SOLUTION 1. Take one of the A's and the B, and choose one at random; call it X (don't look at it). Then choose one at random from X and the other A.

QUESTION 2. Can one do it for p = 4/5 with two A-ballots and one B-ballot only ?


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