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.
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.
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.