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A study considered risk factors for HIV infection among intravenous drug users [11] . It found that \(40 \%\) of users who had \(\leq 100\) injections per month (light users) and \(55 \%\) of users who had \(>100\) injections per month (heavy users) were HIV positive. What is the probability that at least 4 of the 20 users are HIV positive?

4.67 The hospital is planning a new emergency-room facility. It wants enough beds in the emergency ward so that for at least \(95 \%\) of normal-pollution days it will not need to turn anyone away. What is the smallest number of beds it should have to satisfy this criterion? The hospital also finds that on high-pollution days the number of admissions is Poisson-distributed with mean \(=4.0\) admissions per day. Answer Problem 4.67 for highpollution days.

A clinical trial was conducted among 178 patients with advanced melanoma (a type of skin cancer) (Schwartzentruber, et al. [18]). There were two treatment groups. Group A received Interleukin-2. Group B received Interleukin-2 plus a vaccine. Six percent of Group A patients and \(16 \%\) of group \(\mathrm{B}\) patients had a complete or partial response to treatment. Suppose we seek to extrapolate the results of the study to a larger group of melanoma patients. If 20 melanoma patients are given Interleukin-2 plus a vaccine, what is the probability that at least 3 of them will have a positive response to treatment? One issue is that some patients experience side effects and have to discontinue treatment. It was estimated that \(19 \%\) of patients receiving Interleukin-2 plus vaccine developed an arrhythmia (irregular heartbeat) and had to discontinue treatment. since the side effect was attributable to the vaccine, assume that the patients would continue to take Interleukin-2 if arrhythmia developed but not the vaccine and that the probability of a positive response to treatment would be the same as group \(A\) (i.e., \(6 \%\) ).