A health maintenance organization (HMO) is cur­ rently treating 10 patients with a deadly bacterial infec­ tion. The best-known antibiotic treatment is being used in these cases, and this treatment is effective 95 percent of the time. If the treatment is not effective, the patient expires.

a. Define a random variable whose outcome repre­ sents the number of patients being treated by the HMO that survive the deadly bacterial infection. What is the range of this random variable? What is the event space for outcomes of this random vari­ able?

b. Define the appropriate probability density function for the random variable you defined in (a). Define the probability set function appropriate for assign­ ing probabilities to events regarding the outcome of the random variable.

c. Using the probability space you defined in (a) and (b), what is the probability that all 10 of the patients survive the infection?

d. What is the probability that no more than two pa­ tients expire?

e. If 50 percent of the patients were to expire, the government would require that the HMO suspend operations, and an investigation into the medical practices of the HMO would be conducted. Provide an argument in defense of the government's actions in this case.