As a certified quality engineer (CQE), I often focused on the defect rates of manufactured products. They either passed or failed—a binary outcome.
I learned quickly that even a small probability of failure would build up quickly when applying a series of operations. For example, I worked as chief CQE on a chemical plant startup that involved several unit operations in the process line—all to a scale never attempted before. It did not go well. By my reckoning afterwards, each of the steps probably had about a 80/20 chance of succeeding. That led to optimism by the engineers in the company who design our plant. Unfortunately, though, multiplying 0.8 repeatedly is not a winning strategy for process improvement (or gambling!).
As we approach the 80th anniversary of D-Day this diabolic nature of binary outcomes takes on a deadly aspect when you consider how many times our warriors were sent into harms way. The odds continually waver as technology ratchets forward on the offense versus defense. This can be assessed statistically with specialized software such a that provided by Stat-Ease with its logistic regression tools. For example, see this harrowing tutorial on surface-to-air missile (SAM) antiaircraft firing.
Thanks to a heads up from statistician Nathan Yau in one of his daily Flowing Data newsletters, I became aware that many people, even highly educated scientists, get confused about a series of unfortunate or fortunate events (to borrow a phrase from Lemony Snicket).
Yau reports that a noted podcaster with a PhD in neuroscience suggested that chances could be summed, thus if your chance of getting pregnant was 20%, you should see a doctor if not successful after 5 tries. It seems that this should be 100% correct (5 x 20), but not so. By my more productive math (lame pun—taking the product, not summation), the probability of pregnancy comes to 67%. The trick is to multiply the chance of not getting pregnant—0.8—5 times over, subtracting this from 1 and then times 100.
If you remain unconvinced, check out the odds via Yau’s entertaining and enlightening simulation for probability of success for repeated attempts at a binomial process.
Enjoy!