A batch of 300 samples of containers for frozen apple juice contains 20 that are

Question

A batch of 300 samples of containers for frozen apple juice contains 20 that are defective. Three samples are selected at random, without replacement from the batch. (a) Identify the sample space S for this experiment. (b) What is the probability that the first one selected is not defective and the second and third samples are defective? (c) What is the probability that the second one selected is not defective given that the first one was defective? (d) What is the probability that the second and third samples are defective knowing that the first one is defective? (e) What is the probability that all three are defective? (f) What is the probability that all three are acceptable?

Explanation / Answer

a)

As there are just 3 samples, then the number of defectives could be at most 3. If X is the number of defectives, then the sample space is

S = {0,1,2,3} [ANSWER]

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b)

There are 280 nondefetive and 20 defective at first. Hence, without replacement,

P = (280/300)*(20/299)*(19/298) = 0.003980457 [ANSWER]

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c)

If the first one was defective, 280 nondefectives out of 299 are left,

P = 280/299 = 0.936454849 [ANSWER]

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d)

If the first one was defective, 19 defectives out of 299 are left, so

P = (19/299)*(18/298) = 0.003838298 [ANSWER]

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e)

Let a success be getting a defective.

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 =    300      
K = number of successes in the population =    20      
n = sample size =    3      
x = number of successes in the sample =    3      
          
Thus,          
          
P(   3   ) =    0.000255887 [ANSWER]

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f)

Hence, none are defective.

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 =    300      
K = number of successes in the population =    20      
n = sample size =    3      
x = number of successes in the sample =    0      
          
Thus,          
          
P(   0   ) =    0.812453144 [ANSWER]

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