Thread: Representing Knowledge in Probability spaces

The problem:

I flipped a fair coin twice, and I can tell you about one of them: It was a head. What are the chances I got two heads?

A common (and I believe incorrect) answer to the problem takes the form of treating both coins as if they are fair coins, and then creating a table like so:

	P(Head) = 1/2	P(Tail) = 1/2
P(Head) = 1/2	1/2 * 1/2 = 1/4	1/2 * 1/2 = 1/4
P(Tail) = 1/2	1/2 * 1/2 = 1/4	N/A

So according to this model, there are 3/4 chances with a head, 1/4 Head-Head and 2/4 Head-Tail. But the probabilities don't add up to 1, because we aren't including the impossible 1/4 Tail-Tail, which should set off some warning bells.

I suggest the knowledge that one of them is a head is being handled incorrectly. I believe that knowing one of them is a head is equivalent to altering the probability of one of the two coins to be Head:1 and Tail:0. Or to put it another way, every run of an experiment would result in "one of them is a head" being a true statement.

With this perspective on what it means to know one of them is a head, we can construct a different probability table, like so:

	P(Head) = 1/2	P(Tail) = 1/2
P(Head) = 1	1 * 1/2 = 1/2	1 * 1/2 = 1/2
P(Tail) = 0	0 * 1/2 = 0	0 * 1/2 = 0