A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. The hospital also finds that on high-pollution days the number of admissions is Poisson distributed with mean = 4.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. Assume that there are 345 normal-pollution days and 20 high-pollution days in a year.

(a) 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? 

(b) Suppose the hospital decides to have four beds, on a random day, what is the probability it needs to turn someone away? 

(c) With the four bed, if, on a day, the hospital turns someone away, what is the probability that that day is a high-pollution day?

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