 # Suppose that the probability of a company supplying a defective product is a and the probability that the supplied product is not defective is b. Before each product supplied is released for further use, it must be screened through a system which detects the potential defectiveness of the product. The probability that this process successfully detects a defective product is x, and the probability that it does not is y. What is the probability that at least one defective product will pass through the system undetected after 20 products have been screened? Randall Booker 2022-09-13 Answered
Binomial probability
Suppose that the probability of a company supplying a defective product is a and the probability that the supplied product is not defective is b. Before each product supplied is released for further use, it must be screened through a system which detects the potential defectiveness of the product. The probability that this process successfully detects a defective product is x, and the probability that it does not is y. What is the probability that at least one defective product will pass through the system undetected after 20 products have been screened?
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Step 1
Call ${A}_{n}$ the event that the nth product is defective and undetected, then $\mathrm{P}\left({A}_{n}\right)=ay$.
Step 2
The event ${B}_{k}$ that at least one defective product will pass through the system undetected after k products have been screened is the union of k events ${A}_{n}$ hence the complement of ${B}_{k}$ is the intersection of their complements. Although this necessary hypothesis is not mentioned, one probably assumes that the events $\left({A}_{n}{\right)}_{n}$ are independent, then the complementary events are i.i.d. with probabilities $1-ay$. Thus, $\mathrm{P}\left({B}_{k}\right)=1-\left(1-ay{\right)}^{k}$

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