Heart failure is due to either natural occurrences 85 or out

Heart failure is due to either natural occurrences (85%) or outside factors (15%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Suppose that 19 patients will visit an emergency room with heart failure. Assume that causes of heart failure between individuals are independent.

(a) What is the probability that three individuals have conditions caused by outside factors?

(b) What is the probability that three or more individuals have conditions caused by outside factors?

(c) What is the mean and standard deviation of the number of individuals with conditions caused by outside factors?

Round your answers to three decimal places (e.g. 98.765).

Solution

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    19      
p = the probability of a success =    0.15      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.242828896 [ANSWER]

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b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    19      
p = the probability of a success =    0.15      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.441320612
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.558679388 [ANSWER]

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c)

u = mean = np =    2.85 [ANSWER]
  
s = standard deviation = sqrt(np(1-p)) =    1.556438242 [ANSWER]