Blood samples from m persons are to be examined to see whether they contain certain antibodies. Another strategy is to divide the m samples into groups of g pieces (where g is a divisor of m). Within a group, a mixture of the g blood samples is examined. If this mixture does not contain any antibodies, then this one blood test is sufficient. (It is assumed that the mixture always contains antibodies as soon as a single sample contains antibodies). If antibodies are found in the mixture, the g blood samples are tested individually and a total of g + 1 blood test is required for the group.

To evaluate this group strategy, we define the event Ai i-th blood sample contains these antibodies and the random variable Xi = 1Ai (indicator variable) and consider X1, . . . Xm as a consequence of Bernoulli-distributed random variables with parameter p. (For this model assumption it is assumed, for example, that the m persons are a random sample from a large population with a proportion p of persons with these Ancients.)

a) With what probability does a mixture of g blood samples contain these antibodies?

b) Let Y be the number of blood tests required by the group strategy. Represent Y as a function of Xi.

c) Calculate the expected value of Y . For fixed m and g the result is is a function of p. Display this function graphically for m = 2000 and g = 25 (for example, using a drawing program). For which values of p is the group strategy worthwhile, especially for g = 25, and generally for any g and m?

d) For which group size is the range of worthwhile values of p maximum?Specify this maximum range of values of p that are worthwhile for this group size

e) Justify the approximation formula for the optimal group size g with very small p.

f)For which g is the expected value of Y minimal? For p = 0.001, p = 0.01 and p = 0.1, calculate the expected savings in blood tests (in percent) compared to the individual strategy

g) Calculate VarY , the variance of Y . For fixed m and g, the result is a function of p

h)Calculate the maximum digit of as a function of p for any g. This maximum digit then depends on g.

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