Question

Project on Probabilities and rules The contingency table below results from cross classifying U.S. hospitals by type and number of beds. Suppose a hospital is chosen at random. Use proper notation when answering the questions 75 or more beds Total 3945 378 40 10 162 4535 6-24 beds 299 17 25-74 beds 1894 121 6138 516 47 General Psychiatric Chronic Tuberculosis Other Total 0 338 2154 7027 Would it be a rare event for a hospital to be a tuberculosis hospital? 6 Are the events general hospital and 6-24 beds independent? Explain.

Answer 1

Ans 5) The total number of hospitals is 7027, and number of tuberculosis hospitals is 11.

Hence, probability of choosing a tuberculosis hospital = 10 / 7027 = 0.0014 = 0.14%

Hence, it is indeed a rare event for a hospital to be a tuberculosis hospital.

Ans 6) A hospital which is in a general hospital, can also be a hospital with 6-24 beds (in fact we have 299 such hospitals as seen in the given table). Hence, the events general hospital and 6-24 beds are not independent!

Now, let's see the quantitative proof too below:

Let A = Event that the chosen hospital is general hospital, B = Events that the chosen hospital has 6-24 beds

Now, P(A) = No. of general hospitals / Total no. of hospitals = 6138 / 7027

P(B) = No. of 6-24 beds hospitals / Total no. of hospitals = 338 / 7027

And, P(A $\bigcap$ B) = P(Chosen hospital is general hospital $\bigcap$ Chosen hospital has 6-24 beds )

= No. of hospitals which are general hospital and have 6-24 beds / Total no. of hospitals

= 299 / 7027

But, P(A)*P(B) = 6138*338 / 7027², which is not equal to P(A $\bigcap$ B)

Hence, P(A) and P(B) are not independent (using multiplication rule of independent events)