That’s not likely to be something you can calculate, and certainly not from the given information. At the very least, you’d want to know the ratio between P(A wins with 60% of the vote | you vote A) and P(A wins with 60% of the vote | you vote B).
For large numbers of voters who are unaffected by your decision, these are likely to be very close to each other, and so the posterior odds are very close to 50% that the coin flip landed heads.
For smaller numbers (e.g. a board meeting) and/or where your decision may influence other people it’s much more complicated. The fact that in the follow-up question you have an enemy who votes against you implies that the vote is not a secret ballot and your vote does influence at least some other people. This means that the posterior distribution needs to be taken over all sorts of social dynamics and situations.
Even so, the posterior probability of heads isn’t likely to be much different from 50% except in very unusual circumstances.
That’s not likely to be something you can calculate, and certainly not from the given information. At the very least, you’d want to know the ratio between P(A wins with 60% of the vote | you vote A) and P(A wins with 60% of the vote | you vote B).
For large numbers of voters who are unaffected by your decision, these are likely to be very close to each other, and so the posterior odds are very close to 50% that the coin flip landed heads.
For smaller numbers (e.g. a board meeting) and/or where your decision may influence other people it’s much more complicated. The fact that in the follow-up question you have an enemy who votes against you implies that the vote is not a secret ballot and your vote does influence at least some other people. This means that the posterior distribution needs to be taken over all sorts of social dynamics and situations.
Even so, the posterior probability of heads isn’t likely to be much different from 50% except in very unusual circumstances.