1A medical school has 3 affiliate hospitals in Brooklyn 4 in

1)A medical school has 3 affiliate hospitals in Brooklyn, 4 in Manhattan, and 9 in the Bronx. If you randomly choose two different hospitals, what is the chance they are in the same borough? What is the chance they are in different boroughs?

2. At a local clinic, Dr Williams sees all patients with diabetes or hypertension, and Dr Gonzlez sees the remaining patients. Over the past 2 years, records show that Dr Gonzlez sees 55% of patients. Records also show that 20% of patients have diabetes and 40% of patients have hypertension. What percentage of hypertensive patients have diabetes?

3. Human leukocyte antigen (HLA) is a protein found on most cells in your body and is used to match a patient with a donor for a bone marrow transplant. The probability that two siblings are an HLA match is 1 out of 4, and about 30% of patients who need a bone marrow transplant have a suitable donor in their family. The remaining 70% of patients must find a match in a registry of bone marrow donors. The probability that two unrelated people are an HLA match is 1 out of 20,000. Suppose a patient needs a bone marrow transplant and has no siblings. What is the estimated chance that the patient finds a suitable donor in a registry of 30,000 donors, none of which are related to the patient? Round to four decimal places.

Solution

Q1.

Given data, 3 affiliate hospitals in Brooklyn, 4 in Manhattan, and 9 in the Bronx
a)
choose two different hospitals, chance they are in the same borough = 2 Brooklyn OR 2 Manhattan OR 2 Bronx = 3 C 2 + 4 C 2 + 9 C 2
Probability of such occurance = ( 3 C 2 + 4 C 2 + 9 C 2 ) / 16 C 2 = (nCr(3,2)+nCr(4,2)+nCr(9,2))/nCr(16,2) = 0.375

b)
chance they are in different boroughs = 1 - P( chance that are from same) = 1 - 0.375 = 0.625

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