49. In the notation of Section 2, consider the problem of testing Ho : P = Po against HI : P = PI' and suppose that known probabilities 'ITo = 'IT and 'lT1 = 1 - 'iT can be assigned to Ho and HI prior to the experiment. (i) The overall probability of an error resulting from the use of a test cp is 'lTEocp( X) + (1 - 'IT) EI[1 - cp( X)] . (ii) The Bayes test minimizing this probability is given by (8) with k = 'lT0/'lT1• (iii) The conditional probability of Hi given X = x, the posteriorprobability of H; is 'IT;Pi( X) 'lToPo(X) + 'lTIPI(X) , and the Bayes test therefore decides in favor of the hypothesis with the larger posterior probability.