Counting Defects: Binomial and Poisson

Defect counting is simply keeping track of what goes wrong on a production line, and the one idea to hold onto is that a defect is a single flaw while a defective unit is one with at least one flaw. The formulas are two models you already met in the intro course, plus bookkeeping: the binomial (how many of $n$ units are bad), the Poisson (how many flaws land on one unit, with average $\lambda$), the yield $e^{-\lambda}$ (the chance a unit has no flaws at all) and DPMO (flaws per million chances, so simple and complex products can be compared). It looks like four formulas, but it's really two old friends and two ratios.

Defects vs defectives. A circuit board has 1,200 solder joints, and it can come off the line with zero, one or several bad ones. Each bad joint is a defect; a board with at least one is defective. The two counts need different models:

• Defectives among $n$ units, each bad with probability $p$: the binomial. It runs pass/fail inspection and lot sampling.
• Defects on a unit (or per square meter of sheet or per kilometer of cable): the Poisson with mean $\lambda$, the defects per unit (DPU).

Where the Poisson comes from. Each of the board's $n = 1200$ joints fails independently with a tiny probability $p$, so the number of bad joints is binomial with $n$ huge and $p$ tiny. Hold $\lambda = np$ fixed and let $n$ grow. The counting factor and the success part give

and the failure part gives

Multiply: $P(X = k) \to \frac{\lambda^ke^{-\lambda}}{k!}$. Only $\lambda$ survives the limit. You don't need $n$ or $p$ separately, just the average number of defects, and that's why DPU is the currency of defect counting.

Yield. The fraction of units with no defect is $P(X = 0) = e^{-\lambda}$. With DPU 0.6, $e^{-0.6} = 54.9\%$ of boards are perfect, not $1 - 0.6 = 40\%$: some boards collect two or three defects, so more boards escape with none. For small $\lambda$ the two agree, since $e^{-\lambda} \approx 1 - \lambda$ is the start of the Taylor series.

Rolled throughput yield. Steps in a row multiply: casting, machining and painting with DPU 0.10, 0.25 and 0.05 give $e^{-0.10}e^{-0.25}e^{-0.05} = e^{-0.40} = 67\%$. DPUs add and yields multiply. The same law runs chip making: a die of area $A$ on a wafer with $D$ killer defects per cm² has yield $e^{-AD}$, so doubling a chip's area more than halves its yield, which is why big chips are expensive.

DPMO divides DPU by the number of places a defect could occur and scales to a million, so a 2,000-joint board and a 50-joint board can be compared fairly.

When is the Poisson approximation good? When $p$ is small, not merely when $n$ is large. The binomial variance $np(1 - p)$ is smaller than the Poisson's $np$ by the factor $1 - p$, and a classical bound says no probability computed with the Poisson is off by more than the smaller of $p$ and $np^2 = \lambda p$. The textbook rules of thumb, $n \ge 20$ with $p \le 0.05$ or $n \ge 100$ with $np \le 10$, are ways of saying "$p$ is small". With $n = 10$ and $p = 0.3$ the Poisson is badly off.

A common mistake. With DPU 0.6, it's tempting to say 40% of units are perfect, since $1 - 0.6 = 0.4$. That only works if no unit ever has more than one defect. Some units collect two or three, which leaves more units with none, so the yield is $e^{-0.6} = 54.9\%$. Two related slips: forcing a binomial onto flaws per square meter (there's no fixed number of trials), and forgetting to rescale $\lambda$ when the area, length or time changes.

These counts drive p and c control charts and acceptance sampling later in the course. The Poisson also counts failures over time, which leads straight to exponential lifetimes.

Electronics plants track DPMO for every solder process and plan rework stations from the Poisson split of 0, 1 and 2+ defects per board. Chip makers predict die yield with $e^{-AD}$ (and refinements of it) before committing to a chip's size, and paper, film and glass lines report flaws per square meter. Receiving inspection, warranty claims per thousand units and network packet errors are binomial or Poisson counts, and the choice between them is the first modeling decision.