One of the assumptions underlying the theory of control charting (see Chapter 16) 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 .05. What is the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly? Answer this question for 25 successive points.

Answer: Step 1 of 1 Let F be the event called ‘Problem’ Let X be the number of failures. P(F) = 0.05 1 P(F ) = 0.95 The probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly P(at least 1 wrong in 10) = P(X1) = 1 - 0.950 = 0.4013 By similar reasoning if we let Y be the number of problems at 25 points P(at least 1 wrong in 25) = P(Y1) = 1 - 0.955 = 0.7226. Therefore, The probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly is 0.4013 and for probability that at least one of 25 successive points is 0.7226.