Problem 1 Sixty percent of adults over 60 have received a vaccine for shingles. The vaccine is not perfect, and sometimes vaccinated people get the disease. Suppose 20% of adults over 60 have had the disease, while 25% of adults over 60 have had neither the vaccine nor the disease. An adult over 60 is randomly selected. a. If the adult is found to have the disease, what is the probability that the adult had been vaccinated? b. Are the events "the adult is found to have had the disease" and "the adult had been vaccinated" independent?