Find the indicated probability. Round to three decimal places.
-A company purchases shipments of machine components and uses this acceptance sampling plan: Randomly select and test 30 components and accept the whole batch if there are fewer than 3 defectives. If a particular
Shipment of thousands of components actually has a 7% rate of defects, what is the probability that this whole
Shipment will be accepted?
A) 0.649
B) 0.535
C) 0.196
D) 0.279

Question

Quizplus · Accepted Answer

First, we need to determine the probability of having fewer than 3 defectives when 30 components are tested. This can be found using the binomial distribution with n=30 and p=0.07 (assuming the given defect rate applies to the specific shipment being tested):
P(X<3) = P(X=0) + P(X=1) + P(X=2) 
= (30 choose 0) * (0.07)^0 * (0.93)^30 + (30 choose 1) * (0.07)^1 * (0.93)^29 + (30 choose 2) * (0.07)^2 * (0.93)^28
≈ 0.6487
Therefore, the probability of accepting the whole batch is the complement of this probability (i.e. the probability of having 3 or more defectives):
P(accept) = 1 - P(X<3) ≈ 0.3513, which rounded to three decimal places is 0.649, the closest answer choice.

Find the Indicated Probability