A quality control engineer tests the quality of produced com

A quality control engineer tests the quality of produced computers. Suppose that 5% of computers have defects, and defects occur independently of each other. Find the probability of exactly 3 defective computers in a shipment of twenty. Find the probability that the engineer has to test at least 5 computers in order to find 2 defective ones.

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 =    20      
p = the probability of a success =    0.05      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.059582148 [answer]

b)

Using a negative binomial table/technology, we see that

P(at most 2 non defects before 2 defects) = 0.01401875

Thus,

P(at least 3 defects before 2 defects) = 1-P(at most 2 non defects before 2 defects) = 1-0.01401875

=0.98598125 [answer]