We left off last time with the statement of Bayes’s Theorem:
P(A | B) = [P(B | A) P(A)] / P( B ).
Let’s walk through this equation.
From symbols to words
First of all, A | B means “A given B”, and P(…) is “Probability of …”. So if the equation is translated straight into words, it reads, “The probability of A given B is equal to the following: the probability of B given A, times the probability of A, all divided by the probability of B.”
To understand what that means, we should break down each of those pieces further. But first, it’s worth reviewing what probabilities are.
In previous posts, I’ve talked about probabilities that are easy to measure precisely, like the probability of drawing certain cards from a deck, as well as more nebulous probabilities, like those of election outcomes and rain. Bayes’s Theorem will work in both types of situations.
Once you’re comfortable with terms like “probability of 1/6” and “42% chance,” consider how you might find the probability that two different events both happen.
For example, let’s say I flip a quarter and a penny. What are the chances of getting a heads on the quarter and a tails on the penny?
The first step is to notice that the two coin flip outcomes are independent.
Saying two events are “independent” means that the two events don’t have anything to do with each other: one event doesn’t make the other one more likely or less likely.
If my sister eats an omelet for breakfast and I wear green sneakers to work today, those events are independent. If the E train unexpectedly runs local and I’m late to work, those events are not independent, because the first event (slow train) increased the likelihood of the second event (I’m late).
If I flip a quarter and it lands on heads, and I flip a penny and it lands on tails, those events are independent.
(That might be self-explanatory, but in case you’re suffering from the gambler’s fallacy, consider this: If I flip a quarter in the air, it spins a bit, then lands on heads or tails, with equal probability of each. If I flip a penny, it spins, and then it, too, lands on heads or tails. Each little piece of metal, spinning in the air, has nothing to do with the other. So getting a heads on the quarter doesn’t make it more likely that I’d get a tails on the penny, nor vice versa.)
You may recall that the probability of flipping a quarter and getting heads is 1/2, or 50%. The probability of flipping a penny and getting tails is 1/2 as well.
To get the probability of a heads on the quarter and a tails on the penny – since both events are independent – just multiply the two:
(1/2) * (1/2) = 1/4.
So, the probability of a heads, then a tails, is 1/4.
There are good, common-sense reasons why “and” is multiplication – maybe I’ll write a post on it sometime. For now, check out the answers here for a few explanations.
In the next post, we’ll talk about what happens when two events are not independent, and see how Bayes’s Theorem plays a role.