Unfair coin

You have a probably-unfair coin, but you have no idea what the probability it will come up heads is. More precisely, you think that P(H) could be anything between 0 and 1 and that all values are equally likely.

You then flip the coin 4 times and it comes up HHTH.

• What do you think the P(H) is now? Ideally produce the entire probability distribution for P(H).
• If that’s too hard, what is the most likely P(H)?