Suppose that it is mown that the probability p of a head when a certain coin is tossed is either 0.3 or 0.4; and that an experimenter must decide which value of p is the correct one after observing the outcome. head or tail, of just a single toss of the coin. Suppose also that the prior probabilities are as follows:
Finally, suppose that the loss from an incorrect decision is 1 unit. and the loss from a correct decision is O. Show that observing the outcome of a single toss is of no value in this problem because the risk of the Bayes decision procedure based on the observation is just as large as the risk of making a Bayes decision without the observation.