A company that produces DVD drives has a 12% defective rate. Let X represent the

Question

A company that produces DVD drives has a 12% defective rate. Let X represent the number of defectives in a random sample of 53 of their drives. What is the probability the sample will contain exactly 3 defective drives? Give your answer to four decimal places. What is the probability the sample will contain more than 3 defective drives? Give your answer to four decimal places. What is the probability the sample will contain less than 3 defective drives? Give your answer to four decimal places. What is the expected number of defective drives in the sample? Give your answer to two decimal places. What is the variance of the number of defective drives in the sample? Give your answer to four decimal places. What is the standard deviation of the number of defective drives in the sample? Give your answer to four decimal places. Each defective drive costs the company 13 dollars. What is the expected cost to the company for the defective drives in the sample? Give your answer t

Explanation / Answer

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 =    53      
p = the probability of a success =    0.12      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.067822764 [ANSWER]

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    53      
p = the probability of a success =    0.12      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   3   ) =    0.106473361
          
Thus, the probability of at least   4   successes is  
          
P(more than   3   ) =    0.893526639 [ANSWER]

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c)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    53      
p = the probability of a success =    0.12      
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   ) =    0.038650597
          
Which is also          
          
P(fewer than   3   ) =    0.038650597 [ANSWER]

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d)

E(x) = n p = 53*0.12 = 6.36 [ANSWER]

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