In a recent article U.S. News and World Report rated pediatric hospitals and pro

Question

In a recent article U.S. News and World Report rated pediatric hospitals and provided data on several characteristics, including daily inpatient volume and nurse-to-patient ratio, for the top 30 hospitals on its list. Suppose we consider a daily inpatient volume of at least 200 to be high and nurse-to-patient of at least 3.0 to be high. Sixteen hospitals had a daily inpatient volume of at least 200, one-third of the hospitals had a nurse-to-patient ratio of at least 3.0, and seven of the hospitals had both a daily inpatient volume of at least 200 and a nurse-to-patient ratio of at least 3.0.

a. Find the probability of a hospital having a daily inpatient volume of at least 200, the probability of a hospital having a nurse-to-patient ratio of at least 3.0, and the probability of a hospital having both a daily inpatient volume of at least 200 and a nurse-to-patient ratio of at least 3.0.

b. What is the probability that a hospital had a daily inpatient volume of at least 200 or a nurse-to-patient ratio of at least 3.0 or both?

c. What is the probability that a hospital had neither a daily inpatient volume of at least 200 nor a nurse –to-patient ratio of at least 3.0?

Explanation / Answer

let A be the event that a hospital have a daily inpatient volume of at least 200,

    B be the event that a hospital having nurse-to-patient ratio of at least 3.0

given that 16 hospitals out of 30 have a daily inpatient volume of at least 200,

                1/3rd hospitals out of 30 having nurse-to-patient ratio of at least 3.0

               seven hospitals have both.

a) so P[a hospital have a daily inpatient volume of at least 200]=P[A]=16/30 [answer]

    so P[ a hospital having nurse-to-patient ratio of at least 3.0]=P[B]=1/3 [answer]

    so P[a hospital have a daily inpatient volume of at least 200 and having nurse-to-patient ratio of at least 3.0]=P[A and B]=7/30 [answer]

b) P[a hospital have a daily inpatient volume of at least 200 or having nurse-to-patient ratio of at least 3.0 or both]=P[A or B]=P[A]+P[B]-P[A and B]=16/30+1/3-7/30=19/30 [answer]

c) P[a hospital had neither a daily inpatient volume of at least 200 nor a nurse –to-patient ratio of at least 3.0]=P[Ac and Bc]=P[A or B]c=1-P[A or B]=1-19/30=11/30 [by de morgan's law]   [answer]

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