One of the assumptions underlying the theory of control charting is that successive plotted points are independent of one another. Each plotted point can signal either that a manufacturing process is operating correctly or that there is some sort of malfunction. Even when a process is running correctly, there is a small probability that a particular point will signal a problem with the process. Suppose that this probability is 0.07. What is the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly? (Round your answer to three decimal places.)

Answer: a) the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly is 0.4013b) the probability that at least one of 40 successive points indicates a problem when in fact the process is operating correctly is 0.8715  Step-by-step explanation:following how the independence multiplication rule works,i.e finding p(A)' which is 1 - p(No problem) because what we need is an intersection not a union so;a) 10 successive pointsprobability (problem) = 0.05probability (No problem) = 0.95required probability = 1 - [probability (No problem)]^10= 1 - (0.95)^10= 1 - 0.5987 = 0.4013the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly is 0.4013b) 40 successive pointsprobability (problem) = 0.05probability (No problem) = 0.95required probability = 1 - [probability (No problem)]^40= 1 - (0.95)^40= 1 - 0.1285= 0.8715the probability that at least one of 40 successive points indicates a problem when in fact the process is operating correctly is 0.8715