Discrete probability distributions are extremely useful in engineering statistics. One particular area where these distributions are commonly used by engineers is quality engineering.

It turns out that if we know the expected defective rate of a process or product, we can make predictions about actual defectives using the Binomial distribution. For example, if we know our defective rate is 15% and we want to produce 100 of something, the Binomial distribution will allow us to predict the probability of different levels of actual defectives that will be seen among those 100. The expected value of 15% tells us that 15 would be the average value over the long run, but it doesn't tell us how many we'll actually see in the next 100 produced. It might be 0, 5, 10, 15, 20, 25, or 30, and the Binomial distribution can predict the probability of each outcome.

It also turns out that if we have a defective process or product, the number of defects in a defective (also known as the defect density) will be predictable using the Poisson distribution. Some defectives will have only 1 defect, while others might have many more. The parameter of interest with the Poisson distribution is the expected arrival rate. If our quality control data tells us that the expected rate of defects in our defective products or processes is 10, then that number is a statement about how many each defective process or product will contain on average in the long run. It's not a prediction of exactly how many defects the next defective will have. The Poisson distribution gives us the ability to predict the number of defects that we'll see in the next defective, and it will be a value ranging from below to above the expected value.

Discuss the way that probability allows you, as an engineer, to more accurately predict and manage quality in processes and products. Without these distributions, we might know that our defective rate is 15% and that the expected defects in a defective are 10. How would we manage quality if that's all we knew? Alternatively, what do we know about the best case and worst case scenarios if we consider that the Binomial distribution governs the number of defectives we'll see, and the Poisson distribution governs the number of defects we'll see in a defective. How would you use this information to inform management about why sometimes we have good quality days, and other times we have bad quality days? How would you explain that sometimes we have lots of defects but very few defects per defective, while other times we have few defectives but each has lots of defects? How do the Binomial and Poisson distributions help you understand what is happening in each scenario? Use specific examples of probabilities in illustrating your explanations.