# Hypothesis Testing – At What Value of p Are You Indifferent Between Action A and Action B?

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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?

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).