Conditional Probability and Back-of-Envelope techniques

One of the great things about being a maths teacher is finding an intuitive way to explain something. Often in class, children will use a method different from mine. That does not make it worse or better, just more intuitive for them. If we find one, then we discuss it as a class and let the students choose. What I love about this excerpt from Steve Strogatz’s column is the way he, despite being a stunning mathematician, is equally happy learning from his students if it helps them intuit how to solve problems.

Perhaps the most pulse-quickening topic of all is “conditional probability” — the probability that some event A happens, given (or “conditional” upon) the occurrence of some other event B. It’s a slippery concept, easily conflated with the probability of B given A. They’re not the same, but you have to concentrate to see why. For example, consider the following word problem.

Before going on vacation for a week, you ask your spacey friend to water your ailing plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30 percent. (a) What’s the chance that your plant will survive the week? (b) If it’s dead when you return, what’s the chance that your friend forgot to water it? (c) If your friend forgot to water it, what’s the chance it’ll be dead when you return?

Although they sound alike, (b) and (c) are not the same. In fact, the problem tells us that the answer to (c) is 90 percent. But how do you combine all the probabilities to get the answer to (b)? Or (a)?

Naturally, the first few semesters I taught this topic, I stuck to the book, inching along, playing it safe. But gradually I began to notice something. A few of my students would avoid using “Bayes’s theorem,” the labyrinthine formula I was teaching them. Instead they would solve the problems by a much easier method.

What these resourceful students kept discovering, year after year, was a better way to think about conditional probability. Their way comports with human intuition instead of confounding it. The trick is to think in terms of “natural frequencies” — simple counts of events — rather than the more abstract notions of percentages, odds, or probabilities. As soon as you make this mental shift, the fog lifts