Final answer:
To design the optimum decision rule and estimate the average earnings, we need to compare the expected payoffs of each decision. Unfortunately, the question does not provide the probability of getting a head when tossing coin 2, so we cannot calculate the expected payoff for coin 2 or make a decision based on it.
Step-by-step explanation:
To design the optimum decision rule, we need to compare the expected payoffs of each decision. Let's denote the event of choosing coin 1 as C1 and the event of choosing coin 2 as C2. We want to maximize our average earnings, so we should choose the coin that gives us the highest expected payoff.
If we choose coin 1 (C1), we have a probability of 0.5 to choose it and a 0.3 probability of getting a head (H). So the expected payoff for coin 1 is (0.5) * ($1) = $0.50.
If we choose coin 2 (C2), we have a probability of 0.5 to choose it and an unknown probability (P(X2=H)) of getting a head. Since we want to maximize our earnings, we should choose coin 2 only if the expected payoff for coin 2 is higher than $0.50.
To estimate the expected payoff for coin 2, we need to know the probability of getting a head when tossing coin 2 (P(X2=H)). Unfortunately, the question does not provide this information, so we cannot calculate the expected payoff for coin 2.
Without knowing the probabilities of getting heads with coin 2, we cannot design the optimum decision rule or estimate the average earnings.