A manufacturing engineer takes 12 samples from a manufacturi

A manufacturing engineer takes 12 samples from a manufacturing process to verify if the production complies with a quality standard. The probability of a product quality defect is 0.04. What is the probability of finding more than 1 of the samples with quality defects? If the production is a total of 25 products, in which only 5 products does not comply with the quality standard, what is the probability that you take a sample of 4 of these products and two ^2) of them does not cumply with the quality standard?

Solution

a)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.04      
x = our critical value of successes =    1      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   1   ) =    0.919064636
          
Thus, the probability of at least   2   successes is  
          
P(more than   1   ) =    0.080935364 [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 =    25      
K = number of successes in the population =    5      
n = sample size =    4      
x = number of successes in the sample =    2      
          
Thus,          
          
P(   2   ) =    0.150197628 [ANSWER]