calculating probability for a maximum a posteriori hypothesis problem

Question

You are given a coin that may or may not be biased. Specifically, you have three hypotheses about the coin:

H1 = "the coin has probability 1=2 of landing heads"
H2 = "the coin has probability 1=3 of landing heads"
H3 = "the coin has probability 2=3 of landing heads"

I am given that the priors are:

P(H1) = 1/3
P(H2) = 1/3
P(H3) = 1/3

I toss the one coin seven times and observe 5 heads.

I want to calculate P(D|H1), P(D|H2), P(D|H3) but I do not even know where to begin. Do I have to use the hypothesis given to calculate this? Or is there some other way of calculating such?

Where D is the observed event that we see 5 heads and two tails.

Answer 1

"Do I have to use the hypothesis given to calculate this?" Yes. $P(D|H1)$ means the probability of the event $D$ conditional on $H1$, so you obtain it by assuming $H1$ and calculating probability of $D$ under that assumption.