[jules]

Say you have a prior probability distribution P(p) for the probability you think the coin is a coin that comes up heads with probability p. Your probability distribution P(p) will probably have a huge peak around p=0.5, but you can choose any prior belief. So P(p) is your opinion about the coin prior to seeing the experiment. Now we can apply Bayes' theorem to compute your opinion P'(p) about the coin after seeing the experiment:

    P'(p) = P(p | H) 
          = P(H | p)*P(p)/P(H) 
          = p*P(p)/integral(P(H | p)*P(p)dp) 
          = p*/E(P) * P(p)

Your belief that the coin has probability p is skewed by a factor of p/E(p).

Here's an example of a graph of P(p) that shows how your belief about the coin is skewed after seeing a heads:

http://dl.dropbox.com/u/388822/coin.png

The first graph is an example of a prior belief about the coin, the second graph is the belief that this person should have after seeing the experiment.

So the answer to the question is:

    P'(1) = 1/E(P) * P(1) = P(1)/E(P)

i.e. your new probability that this is a coin that always comes up heads is your old probability divided by your expected value of the probability of coming up heads.

For example if your prior belief was unbiased, then E(P)=0.5, and P'(1) = 2*P(1).

[Jabbles]

Your prior belief that the coin is "special" should be extremely small, since you've examined it and can see no reason for it to come up heads all the time. After flipping, your belief will be larger, but still small.

[jules]

Any prior belief is valid. That's the point of this: you separate the mathematical reasoning from the subjective assumptions (beliefs). For example it might be the case that Tom has demonstrated several of those special coins before, and in that case your opinion would probably be that there is a good chance that this one is special too. The nice thing about the math is that we can encapsulate these assumptions in the prior probability distribution P(p).

BTW I used that distribution in the plots because it was easiest to come up with, and somewhat realistic, and it shows the skewing well. Feel free to plug in your own beliefs.