Problem 5. (20 points) A blood test is being performed on n individuals. Each person can be tested separately, but this is expensive. Pooling can decrease the cost. The blood samples of k people can be pooled and analyzed together. If the test is negative, this one test suffices for the group of k individuals. If the test is positive, then each of the k persons must be tested separately and thus k +1 total tests are required for the k people. Suppose we create n/k disjoint groups of k people (where k divides n) and use the pooling method. Assume that each person has a positive result on the test independently with probability p. (a) What is the probability that the test for a pooled sample of k people will be positive? (b) What is the expected number of tests necessary? (c) Describe how to find the best value of k. Problem 5. (20 points) A blood test is being performed on n individuals. Each person can be tested separately, but this is expensive. Pooling can decrease the cost. The blood samples of k people can be pooled and analyzed together. If the test is negative, this one test suffices for the group of k individuals. If the test is positive, then each of the k persons must be tested separately and thus k +1 total tests are required for the k people. Suppose we create n/k disjoint groups of k people (where k divides n) and use the pooling method. Assume that each person has a positive result on the test independently with probability p. (a) What is the probability that the test for a pooled sample of k people will be positive? (b) What is the expected number of tests necessary? (c) Describe how to find the best value of k