Answer: To calculate the probability that the shipment is rejected, we need to find the probability that at least one of the three components selected is defective.
First, we can find the probability that none of the three components selected is defective, which would mean the shipment is accepted. To do this, we use the hypergeometric distribution formula:
P(X = k) = (C(N-k,n-K) * C(K,k)) / C(N,n)
where:
N = total number of components in the shipment (160)
K = number of defective components in the shipment (4)
n = number of components selected for testing (3)
k = number of non-defective components among the selected components (i.e., all 3 components are non-defective)
Plugging in the values, we get:
P(all 3 components are non-defective) = (C(160-4-3,3-0) * C(4,0)) / C(160,3)
= (C(153,3) * C(4,0)) / C(160,3)
= (22,446 * 1) / 6,384,320
= 0.00352
Therefore, the probability that the shipment is accepted is 0.00352.
To find the probability that the shipment is rejected, we can subtract this probability from 1, since the probabilities of all possible outcomes must add up to 1:
P(at least one component is defective) = 1 - P(all 3 components are non-defective)
= 1 - 0.00352
= 0.99648
Therefore, the probability that the shipment is rejected is 0.99648, or approximately 99.65%.
Explanation: