Question 4

Scenario: 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.

Events:

  • - A process is actually running correctly.
  • - A process signals that it's running correctly.
  • - A process signals that there's a problem.

Probabilities:

  • .

(a)

Problem: What is the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly?

Process: The question can be rephrased to ask what the probability is of not all points signaling correctly. The situation which makes the statement "all points are signaling correctly" false is if at least one point is signaling incorrectly. Since the points are independent in this scenario, raising any probability related to those points by a power equivalent to the number of points being checked is how you find the probability that those points will be in the same state. So, the probability that ten points are incorrectly signaling is equivalent to the complement of the probability of a process signaling correctly raised to the power of the number of points being checked. In this case, the number of points being checked is 10.

Answer: 0.461.

(b)

Problem: What is the probability that at least one of 40 successive points indicates a problem when in fact the process is operating correctly?

Process: This is the same situation as the previous problem, except more points are being checked. Instead of raising the probability of a process signaling correctly to the power of 10, we're raising it to the power of 40.

Answer: 0.916.