I like the idea of having pictures but I do not like the idea of procuring pictures. I'll make it a higher priority for future posts, though, and if someone wants to send me pictures (which I can legally use) for this post I'll be happy to edit them in.

I replaced the "x"s with "p"s; hopefully that'll make it a bit clearer.

We start off with a prior P(p)=1. That is, I think every p is equally likely, and when I integrate over the domain of p (from 0 to 1) I get 1, like I should.

Then I update on seeing heads. For each p value, the chance I saw heads was p- and so I expect my function to have the functional form P(p)=p. Notice that after seeing heads I think the mode is a coin that always lands on heads and that it's impossible that the coin always lands on tails- both are what I expect. When I integrate p from 0 to 1, though, I get 1/2. I need to multiply it by 2 to normalize it, and so we have P(p)=2p.

This might look odd at first because it sounds like the probability of the coin always landing on heads is 2, which suggests an ill-formed probability. That's the probability density, though- right now, my prior puts 0 probability on the coin always landing on heads, because that's an integral with 0 width.

The 2-2x comes from the same argument, but the form is now 1-x.