Twocombinationdrugtherapies(TreatmentAandTreatmentB)have beendeveloped for eradicating Helicobacter pylori in humanpatients. The effectiveness of these treatments depends on whetheror not the patient is resistant to the chemical compoundMetronidazole, but apatient’s resistance status is not routinelydetermined before beginningtreatment. Treatment A successfullyeradicates Helicobacter pylori in 92% of resistantpatients and 87% of non-resistant patients. The correspondingproportions for Treatment B are 75% and 95%.
(i) Denote by θ (0 < θ < 1) the proportion of the affectedpopulation that is resistant. If a patient from this population isunsuccessfully treated with Treatment B, write down an expressionfor the probability that the patient is resistant.
(ii) For what values of θ would a greater proportion of patientsfrom this population be successfully treated by Treatment B than byTreatment A?
(iii) Suppose that θ = 0.2. If 20 patients, selected at randomfrom the affected population, are independently treated withTreatment B, find the probability that at least 18 of them will betreated successfully.
