A confused-looking man on the street asks if you want to play a game of chance. The rules are simple: He flips his coin, and if it lands on heads, you get $1. If it lands on tails, you get nothing. His coin, however, is very strangely shaped - it's bent all over, and you have no idea if it's even remotely fair. The man flips the coin once, and it lands on heads.

You can assume that you know nothing about the true probability of heads for the coin before you saw the first head, and that probability is not going to change over time.

What is the maximum price you should be willing to pay to play this game now?

To further generalize, assuming any prior sequence of flips for the coin, what price is it worth to you to play the game?

Also, what if the true probability of heads can only be one of a discrete set of values?

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