If 14 of disks produced on an assembly are defective whats t

If 14% of disks produced on an assembly are defective, what\'s the probability that there will be exactly 2 defects in a random sample of 20? 4 defects? Less than 6 defects?

Please break problem down without Excel.

Solution

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 =    0.14      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2 defects) =    0.246593596

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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 =    0.14      
x = the number of successes =    4      
          
Thus, the probability is          
          
P (    4 defects) =    0.166640724

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Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.14      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.949327261
          
Which is also          
          
P(fewer than   6   ) =    0.949327261 [ANSWER]