As you likely know, the Poisson distribution is an approximation of the binomial. In general form, the Poisson function is given as Y = (np^r * e^-np)/r!, where n is the number of trials, p is the event probability, and r is the number of event occurrences. By direct substitution, we assert that Y = (dpu^r * e^-dpu) / r!, where dpu is the classic defects-per-unit quality metric. Thus, the expected number of units (n) containing exactly r defects can be given as n = Y * u.

For example, consider the case u = 500 and dpu = 1.0. Based on these facts, we would compute the probability of experiencing exactly zero defects (per unit) to be approximately Y = 0.367879, or about 36.8 percent. Of course, this can also be viewed as “throughput yield.” Given this probability, we would anticipate that n = 184 of the u = 500 production units would yield upon completion of the process. In other words, one could expect 184 out of 500 units to contain exactly zero defects. As yet another example, we might choose to compute the probability of exactly 4 defects (per unit). Doing so would reveal the probability to be about 1.5 percent. This would mean that we could expect 8 out of 500 units to contain exactly 4 defects.