12.37 A mixed group: probabilities. We would like to find the probability that exactly 2 of the 20 exposed children in the previous exercise develop whooping cough.

(a) One way to get 2 infections is to get 1 among the 17 vacci nated children and 1 among the 3 unvaccinated children. Find the probability of exactly 1 infection among the 17 vaccinated children. Find the probability of exactly 1 infection among the 3 unvaccinated children. These events are independent: what is the probability of exactly 1 infection in each group?

(b) Write down all the ways in which 2 infections can be di vided between the two groups of children. Follow the pattern of part

(a) to find the probability of each of these possibili ties. Add all of your results (including the result of part (a))
to obtain the probability of exactly 2 infections among the 20 children.