The Probability of Reoccurrence: P (r)

by Dennis R. Owens

Bob, the quality manager at Acme Manufacturing Co., was sent a supplier corrective action request (SCAR) from Acme’s largest customer, Build-Rite Industries. Bob assembled a team of engineers to identify the problem and determine a root cause.

While glancing at the SCAR form, Bob noticed a request for assurance through the calculation of the probability of reoccurrence. Furthermore, the SCAR stated that any reoccurrence probability greater than 10% would be cause for rejection and all future orders would be placed on hold.

In a panic, Bob called Greg, the company statistician, and the team to a conference room and explained his dilemma. After hearing Bob’s concern, Greg told the team, “I think I know a simple method to solve this problem. We need to use a process approach and calculate the probability of a reoccurrence.”

“What is probability of a reoccurrence?” asked Bob.

“Every problem—or conversely, the opportunity to perform a task—is a process,” Greg said. “We know all processes can be affected by six basic components: manpower, machines, methods, materials, measurements and Mother Nature (the environment)—otherwise known in quality assurance circles as the ‘cause and effect’ diagram.”

“There are three basic principles that we must assumed in order to correctly interpret the results of this type of approach,” Greg said.

  • Principle one: The cause and effect diagram is a holistic approach to causal analysis describing all contributing factors surrounding a problem or opportunity for performance.
  • Principle two: Each event or factor (“M”) is independent of the others.
  • Principle three: Only events or factors that could lead to a direct output P (r) failure are used in the analysis. For example, assume a failure in one event or factor will not cause a failure in one of the other applicable events.

To prove his point, Greg offered an example. “Let’s say we could collect data that strongly suggested the probability of each independent reoccurrence event or factor solution was 5%. Based solely on this information, Bob, should we submit the SCAR to the customer?”

“Well sure,” Bob answered.” That sounds pretty good to me.”

“Hold on, Bob,” Greg said. “Let’s take a further look to see what this really means.”

Using probabilities based on variable data and the product rule for independent events, Greg arrived at the following equation:

P (r) = 1 - (pi x p2 x p3.…pn)

in which:

P (r) = Probability of an reoccurrence

pi = Prob (event Mi occurs) = prob (aspect i of problem is solved)

n = Number of event factors (or M factors)

i = 1 to 6 (applicable independent events factors)

Greg then offered the following sample problem for Bob with specific figures:

Pi = 0.95 (95%) probability for each event factor

n = 6 (all applicable event factors for a particular analysis)

Greg derived the following answer:

P (r) = 1 – (0.95 x 0.95 x 0.95 x 0.95 x 0.95 x 0.95) = 1 – 0 .735 = 0.265 or 26.5%.

This amounts to a 26.5% probability (or chance) this failure will happen again. “Therefore, we have not met the customer’s acceptance threshold and would not submit the SCAR without re-evaluation of our solutions,” Greg said.

Shocked by this result, Bob asked: “What do we do if the probabilities are not equal?”

“That’s OK,” Greg replied. “The output of P (r) is the product of all the applicable event or factor probabilities. Whether you have six factors or two, the process is still a valid way of predicting vulnerabilities and reoccurring nonconformity.”

“Now what?” Bob asked.

Greg suggested the following action:

  1. Start a causal analysis exercise. Using the 6M approach (Figure 1) is a good start, but it is only a generic model. So the team must assign the branches commensurate with the situation.
  2. Identify independent events or factors that are applicable.
  3. Decide what to solve. Remember the economic or business considerations that will affect choices.
  4. Derive the individual probabilities of reoccurrence based on the team’s experience and judgment of the situation. Screen each event or factor, and determine whether it is a contributor to the problem.
  5. Calculate the probability of reoccurrence value from the equation.

Bob seemed somewhat relieved. He was a bit familiar with causal analysis and preventive action but never thought a quantifiable determination could be derived from these quality concepts. Later, he visited Greg.

“Greg, the customer accepted our response. I really learned something today,” Bob said. “We really need to step up our game when it comes to preventive action. You’ve shown us the way. I’m confident to use this approach with our senior management and give them real quantitative data to back up how successful we are when fixing problems.”

“Well Bob,” replied Greg. “I think you’ve just stepped up the game.”

DENNIS OWENS is the senior member of technical staff at Sandia National Laboratories in Albuquerque, NM. He earned his master’s degree in organizational management at the University of Phoenix in Albuquerque. Owens is a certified lean Six Sigma Black Belt and a certified quality systems provisional auditor. He is a member of ASQ.

Good use of Simple method of cause and effect diagram for determining Probability of reoccurrence.
--Barge N J, 06-12-2009

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