9. (i) As an example in which randomization reduces the maximum risk, suppose that a coin is known to be either standard (HT) or to have heads on both sides (HH). The nature of the coin is to be decided on the basis of a single toss, the loss being 1 for an incorrect decision and 0 for a correct one. Let the decision be HT when T is observed, whereas in the contrary case the decision is made at random, with probability p for HT and 1 -p for HH. Then the maximum risk is minimized for p = . (ii) A genetic setting in which such a problem might arise is that of a couple, of which the husband is either dominant homozygous (AA) or hetero- zygous (Aa) with respect to a certain characteristic, and the wife is homozygous recessive (aa). Their child is heterozygous, and it is of importance to determine to which genetic type the husband belongs. However, in such cases an a priori probability is usually available for the two possibilities. One is then dealing with a Bayes problem, and randomi- zation is no longer required. In fact, if the a priori probability is p that the husband is dominant, then the Bayes procedure classifies him as such if p> and takes the contrary decision if p