{a} Let's consider a single group. Suppose. in a pooled test. that a group consists of specimens from It people. Write expressions for the following probabilities in terms of p and k. - The probability that none of the In people has the disease. . The probability that at least one of the it people has the disease. According to the pooling method, if none ofthe it people has the disease, only one test will be I'LII'I for that group; if at least one of them has the disease, than after the initial test on the mixed specimen, the saved specimens from each of the it people will be tested. so in all k + 1 tests will be run for the group. Let E be the random variable representing the number of tests that are run for the group. Write an expression. in tem'ls of p and k. for the expected value of E. {b} Now let's consider the entire community. Suppose there are N people in the community, and they are screened for the disease using the pooling method. Blood samples are taken from each person and the samples are divided into groups of size It. This means there will be M k groups of samples. each of size k. {For convenience. we're assuming that N is divisible by k.) We'll call the groups Group 1. Group 2. . . . . Group N k. LetE1. 22. . . . . E N k be random variables representing the number oftests run for Group 1. Group 2. . . . . Group N k. respectively. Notice that the distribution of each Zi is exame the same as that of the random variable I in Question 2a. Also let Tk be the random variable representing the total number of tests run for the community of N people when pooling is done with groups of k. people. You'll agree that Tk=z1+32+- - -+?_le. Recall that our goal is to nd an expression for the expected value of T. We have seen in class that expectation is a linear operation. which means that \"the expectation of a sum of random variables is the sum of the expectations of the random variables\". Hence E{Tlr.}= E1121} + E{?_2] + . . . + E{E Milt]. Write an expression, in terms of p and k. for the expected value of T