Question

A company purchases shipments of machine components and uses this acceptance sampling plan: Randomly select and test 26 components and accept the whole batch if there are fewer than 3 defectives. If a particular shipment of thousands of components actually has a 6% rate of defects, what is the probability that this whole shipment will be accepted?

Answer 1

Answer: 0.7973

Explanation:

Binomial probability formula :-

$P(x)=^nC_x\ p^x(1-p)^{n-x}$, where P(x) is the probability of getting success in x trials , p is the probability of success in one trial and n is the number of trials.

Given : The probability of getting a defect components : $0.06$

If randomly select and test 26 components , then the probability that this whole shipment will be accepted will be :-

$P(x<3)=P(0)+P(1)+P(2)\\\\=^{26}C_{0}(0.06)^0(0.94)^{26}+^{26}C_{1}(0.06)^1(0.94)^{25}+^{26}C_{2}(0.06)^2(0.94)^{24}\\\\=(0.94)^{26}+26(0.06)(0.94)^{25}+325(0.06)^2(0.94)^{24}\approx0.7973$

Hence, the  probability that this whole shipment will be accepted = 0.7973