A die is weighted so that when it is tossed, one particular number will come up with a probability of 1/2 and each of the five other numbers will come up with a probability of 1/10. Suppose that each of the six sides has had an equal chance of being the side that is favored. We are interested in deciding which is the favored number.

(a) With no observations, what is an optimal decision rule and what is the probability of error?

(b) Suppose we observe a 3 and a 4 on two independent rolls of the die. Given this data, what is an optimal decision rule and what is the probability of error?