A sample of 400 large companies showed that 130 of them offer free health fitness centers to their employees on the company premises. If one company is selected at random from this sample, what is the probability that this company offers a free health fitness center to its employees on the company premises? What is the probability that this company does not offer a free health fitness center to its employees on the company premises? Do these two probabilities add up to 1.0? If yes, why?

One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 90$\%$ detection rate for carriers and a 5$\%$ detection rate for noncarriers. Suppose the test is applied independently to two different blood samples from the same randomly selected individual. (a) What is the probability that both tests yield the same result? (b) If both tests are positive, what is the probability that the selected individual is a carrier?