You are in charge of coffee bean quality control. In a very large batch of coffee beans, you estimate that the chance a bean is roasted incorrectly is 0.008. You have your employees sample 1000 beans from the seemingly endless supply, to see how many incorrectly roasted beans they find. (The roast of the beans is assumed to be independent, since the quantity of beans is large, and the beans are well-mixed before they get inspected.)
a. What is the expected number of incorrectly roasted beans in the 1000-item sample? What is the standard deviation?
b. Set up the exact probability (but do not solve) that at least 2 beans from the sample are incorrectly roasted.
c. What is the approximate probability that at least 2 beans from the sample are incorrectly roasted?