A blood bank has available 10 units of A+ blood. In fact, four are contaminated

Question

A blood bank has available 10 units of A+ blood. In fact, four are contaminated with serum hepatitis.

a. Three units are randomly selected for use with three patients. How many different ways can we select three units of blood from the 10 available unit?

b. Now we are going to give these blood units to three different patients. What is the probability that exactly one of these patients will be exposed to hepatitis from this source?

c. What is the probability that at least two of the three will be exposed to hepatitis?

Explanation / Answer

A)

There are 10C3 = 10!/(3!7!) = 120 ways [ANSWER]

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b)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    10      
K = number of successes in the population =    4      
n = sample size =    3      
x = number of successes in the sample =    1      
          
Thus,          
          
P(   1   ) =    0.5 [ANSWER]

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c)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative hypergeometric distribution table or technology, matching          
          
where          
N = population size =    10      
K = number of successes in the population =    4      
n = sample size =    3      
x = critical number of successes in the sample =    2      
          
Thus,          
P(at most   1   ) =    0.666666667
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.333333333 [ANSWER]

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