2 The drug EPO (erythropoetin) is taken by some athletes to improve their performance. This drug is in fact banned and blood samples taken from athletes are tested to measure their 'hematocrit level'. If the level is over 50 it is considered that the athlete is likely to have taken EPO and the result is described as 'positive'. The measured hematocrit level of each athlete varies over time, even if EPO has not been taken.

1. For each athlete in a large population of innocent athletes, the variation in measured hematocrit level is described by the Normal distribution with mean 42.0 and standard deviation 3.0.
   (A) Show that the probability that such an athlete tests positive for EPO in a randomly chosen test is 0.0038 .
   (B) Find the probability that such an athlete tests positive on at least 1 of the 7 occasions during the year when hematocrit level is measured. (These occasions are spread at random through the year and all test results are assumed to be independent.)
   (C) It is standard policy to apply a penalty after testing positive. Comment briefly on this policy in the light of your answer to part (i)(B).
2. Suppose that 1000 tests are carried out on innocent athletes whose variation in measured hematocrit level is as described in part (i). It may be assumed that the probability of a positive result in each test is 0.0038 , independently of all other test results.
   (A) State the exact distribution of the number of positive tests.
   (B) Use a suitable approximating distribution to find the probability that at least 10 tests are positive.
3. Because of genetic factors, a particular innocent athlete has an abnormally high natural hematocrit level. This athlete's measured level is Normally distributed with mean 48.0 and standard deviation 2.0. The usual limit of 50 for a positive test is to be altered for this athlete to a higher value \(h\). Find the value of \(h\) for which this athlete would test positive on average just once in 200 occasions.