Sandsynlighedsregning - Bayes rule

An experimenter observes the occurrence of an event A as the resultof a particular experiment. There are three different hypotheses,H1, H2, and H3, which the experimenter regards as the only possibleexplanations of the occurrence of A. under hypothesis H1, theexperiment should produce the result A about 10% of the time overthe long run, under H2 about 1% of the time, and under H3 about 39%of the time. Having observed A, the experimenter decides that H3 isthe most likely explanation, and that the probability that H3 istrue is

39 %/( 10%+1%+39%) =78%

a)What assumption is the experimenter implicitly making?

b)Does the probability 78% admit a long-run frequencyinterpretation?

c)Suppose the experiment is a laboratory test on a blood samplefrom an individual chosen at random from a particular population.The hypothesis Hi is that theindividual’s blood is of some particular type i. over the whole population it is known thatthe proportion of individuals with blood of type 1 is 50%, theproportion with type 2 blood is 45%, and the remaining proportionis type 3. Revise the experimenter’s calculation of theprobability of H3 given A, so that is admits a long-run frequencyinterpretation. Is H3 still the most likely hypothesis given A?