Chapter 12.7, Problem 61E

Probability of at least one encounter Suppose that in a large group of people, a fraction 0 ≤ r ≤ 1 of the people have flu. The probability that in n random encounters you will meet at least one person with flu is P = f(n, r) = 1 – (1 – r)ⁿ. Although n is a positive integer, regard it as a positive real number.

a.    Compute f_r and f_n.

b.    How sensitive is the probability P to the flu rate r? Suppose you meet n = 20 people. Approximately how much does the probability P increase if the flu rate increases from r = 0.1 to r = 0 11 (with n fixed)?

c.    Approximately how much does the probability P increase if the flu rate increases from r = 0.9 to r = 0.91 with n = 20?

d.    Interpret the results of parts (b) and (c).