At what value of p are you indifferent between action A and action B? Problem statement: Suppose you are deciding between two actions, A, and B, and are testing between two mutually exclusive hypotheses, H1 and H2. If you choose action A, you receive 1 dollar if H1 is true and nothing if it is false.  If you choose action B, you receive 2 dollars if H1 is true and lose 1 dollar if it is false. Suppose H1 is true with posterior probability p.
Question: At what value of p are you indifferent between action A and action B?
I understand the problem statement but i don't understand the question and I have no idea where I should start? How can I solve this problem?
 A: Let's look at this problem in terms of utility.  Let $U(x \vert \theta)$ be the utility under a given action, $x=A, B$ and state of the world/hypothesis $\theta = H_1, H_2$.
Because there are two possible hypotheses and 2 actions, we can enumerate our utilities
$$ U(A \vert H_1) = 1 $$
$$ U(A \vert H_2) = 0 $$
$$ U(B \vert H_1) = 2 $$
$$ U(B \vert H_1) = -1 $$
We are also given the information about the posterior probability of one of the hypotheses, $P(H_1) = p$.  Since the hypotheses are mutually exclusive, $P(H_2) = 1-P(H_1)$. Using this information, we can compute the expected utility for each action, with I will denote $E(U(x))$.
$$E(U(A)) = U(A\vert H_1)P(H_1) + U(A \vert H_2)P(H_2) = p$$
$$E(U(B)) = U(B\vert H_1)P(H_1) + U(B \vert H_2)P(H_2) = 2p - (1-p) = 3p-1$$
You're indifferent to the decision when each decision has the same expected utility.  So solve $3p-1=p$ or $p=0.5$.  This makes sense, when there is an equal chance of either hypothesis being true then both actions have the same expected pay off (namely a dollar).
