ONE GOOD IDEA
Try a decision tree to select the right discrete distribution
by Edwin G. Landauer
It can be difficult to identify the appropriate discrete distribution to use when attempting to determine probabilities in a particular situation.
Whether in real life or for exam purposes (such as for the certified quality engineer exam), the decision tree in Figure 1 can help you determine the appropriate distribution.
Let’s use the following sample test questions to demonstrate the use of the decision tree:
Question one: A process is producing material that is 40% nonconforming. Four pieces are selected at random for inspection. What is the probability of exactly one good piece being found in the sample?
Solution: Here, we have count data (the number of good pieces), and for each piece, there are two outcomes (conforming and nonconforming). We also have independent trials—that is, four pieces selected at random from a large population, and there is a fixed number of trials (four pieces selected). This indicates we have a binomial distribution.
Question two: Suppose six bad parts get mixed up with nine good parts. If two parts are drawn simultaneously, what is the probability both are good?
Solution: Here, we have count data (the number of good parts), and for each part, there are two outcomes (good and bad). We also have dependent trials—that is, a small set of parts, and the second draw depends on the first. This indicates we have a hypergeometric distribution.
Question three: An inspection plan is set up to randomly sample three feet of a 100-foot cable and accept the cable if there are no flaws found in the three-foot length. What is the probability a cable with an average of one flaw per foot will be rejected by the plan?
Solution: Here, we have count data (the number of flaws). We are counting the number of occurrences per unit (flaws per three feet of cable length), and the occurrences are independent—that is, one occurrence does not influence the next occurrence. This indicates we have a Poisson distribution.
Question four: In a certain manufacturing process, it is known that an average of one in every 100 items is nonconforming. What is the probability the fifth item inspected is the first nonconforming item found?
Solution: Here, we have count data (the number of items inspected), and there are two outcomes per item (conforming and nonconforming). We also have independent trials—that is, items selected at random from a large population—and an unknown number of trials that continue until a nonconforming item is found. This indicates we have a geometric distribution.
Question five: Suppose that 60% of the parts produced by a particular machine are acceptable, 30% need to be reworked to be acceptable, and 10% are unacceptable and will be scrapped. What is the probability that among 15 parts randomly selected for audit, nine are acceptable, three will need to be reworked, and three are scrapped?
Solution: Here we have count data (number of parts produced), and there are more than two outcomes per trial (acceptable, reworked and scrapped). We also have independent trials—that is, parts selected at random from large population. This indicates we have a multinomial distribution.
Improving proficiency, accuracy
As these examples demonstrate, with a little practice, you can become proficient at interpreting the test questions involving discrete distributions. The use of the decision tree can optimize test time and improve the accuracy of the answers.
Edwin G. Landauer is a retired professional engineer and college instructor. He has master’s degrees in industrial engineering from the University of Central Florida in Orlando, in statistics from Montana State University in Bozeman, and in mathematics from Portland State University in Oregon. Landauer is an ASQ fellow and an ASQ-certified quality engineer.