A quality control inspector at an electronics factory tests components to see if any are defective. The inspector randomly selects \(30\) components for testing in a day. Historical data shows that \(0.8\%\) of the components are defective.
State, in context, two assumptions needed for the number of defective components found each day to be well-modelled by a binomial distribution.[2]
Assume now that the number of defective components found each day has a binomial distribution.
Find the probability that at most \(1\) component is found to be defective.[1]
Find the probability that the first defective component is the fourth component tested and a total of \(3\) defective components are found in a day.[2]
Given that the inspector finds at least \(1\) defective component in a day, find the probability that she finds less than \(3\) defective components.[2]
Find the probability that in a randomly chosen \(5\)-day working week, at most \(1\) component is found to be defective on no more than \(3\) days.[2]
The owner of the factory decides to expand his business by producing memory chips. On average, \(5\%\) of the memory chips are defective. Another quality control inspector was tasked to randomly select \(100\) memory chips for testing in a day. You may assume that the number of defective memory chips found each day has a binomial distribution.
The inspector records down the number of defective memory chips she found each day. Estimate the probability that the average number of defective memory chips found on \(50\) randomly chosen days is between \(4.5\) and \(5.5\).[3]