SOLUTION: The probability that a certain machine turns out a defective item is .05. Find the probabilities that in a run of 75 items, the following results are obtained. a. Exactly 5 defe

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Question 697630: The probability that a certain machine turns out a defective item is .05. Find the probabilities that in a run of 75 items, the following results are obtained.
a. Exactly 5 defective items
b. No defective items
c. At least 1 defective item
Answer by Positive_EV(69) About Me  (Show Source):
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The distribution of the number of defective items in this run is binomial with n = 75 items and p = .05 of each item being defective. The probability mass function of the binomial distribution is:

P%28X+=+k%29+=+%28nCk%29%2A%28p%5Ek%29%2A%281-p%29%5E%28n-k%29, where nCk is the number of combinations of k objects chosen from n = n%21%2F%28k%21%2A%28n-k%29%21%29

1) In this case, we set k = 5. n = 75 and p = .05, so:

P%28X+=+5%29+=+%2875C5%29%2A%28.05%5E5%29%2A%28.95%29%5E%2870%29
P%28X+=+5%29+=+%2875%21%2F%285%21%2A%2870%29%21%29%29%2A%28.05%5E5%29%2A%28.95%29%5E%2870%29


2) In this case, we set k = 0. The binomial probability mass function reduces to %281-p%29%5En when k = 0, so P(X = 0) = %28.95%29%5E75+=+.0213

3) The event that at least one item is defective is the compliment of the event that there are no defective items. The probability of a complimentary event happening is 1 - P(original event), so the probability of at least one defective item is 1 - .0213 = .9787.