Question
A new integrated circuit board is being developed for use in computers. In the early stages of development, a lack of quality control results in a 0.2 probability that a manufactured integrated circuit board has no defects. Engineers need 24 integrated circuit boards for further testing. What is the minimum number of integrated circuit boards that must be manufactured in order to be at least $98 \%$ sure that there are at least 24 that have no defects?
A new integrated circuit board is being developed for use in computers. In the early stages of development, a lack of quality control results in a 0.2 probability that a manufactured integrated circuit board has no defects. Engineers need 24 integrated circuit boards for further testing. What is the minimum number of integrated circuit boards that must be manufactured in order to be at least $98 \%$ sure that there are at least 24 that have no defects?
Answers
Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives computer boards in lots of five. Two boards are selected from each lot for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair $(1,2)$ represents the selection of boards 1 and 2 for inspection.
(a) List the ten different possible outcomes.
(b) Suppose that boards 1 and 2 are the only defective boards in a lot of five. Two boards are to be chosen at random. Define $X$ to be the number of defective boards observed among those inspected. Find the probability distribution of $X .$
(c) Let $F(x)$ denote the cdf of $X .$ First determine $F(0)=P(X \leq 0), F(1),$ and $F(2),$ and then obtain $F(x)$ for all other $x .$
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