# Bayes’s Theorem: Part I

On a Friday afternoon in 1989, the physicist Leonard Mlodinow got a phone call from his doctor. His doctor told him there was an over 99 percent chance that he was infected with HIV.

The doctor was wrong. Not only was the physicist HIV-free, the chances that he was sick had really been less than 10 percent.

The doctor had done his medical testing perfectly. He just messed up the statistics.

In particular, he failed to use Bayes’s Theorem.

The next few posts will be about Bayes’s Theorem, which is critically important in probability and statistics – and, in my opinion, for thinking rationally in general.

We’ve talked about probability in previous posts. Statistics also deals with prediction – it’s the math used for dealing with data. “Data” here refers to big collections of  information. So, for example, you might have data about a baseball player’s batting average, the trends of a tech company’s stock, or the algae growth in a pond. With respect to the examples in these next few posts, probability and statistics are essentially two sides of one coin.

My fifteen-year-old brother loves birdwatching. Recently, outside our house, we saw a black bird with dull orange feathers on its stomach.

My brother said the bird could be either a Baltimore oriole or an orchard oriole. Those are two similar-looking, but different, types of birds. Baltimore orioles are usually black and bright orange; orchard orioles are usually black and dull orange.

The bird we saw had the coloring of an orchard oriole. Normally, I’d just identify the bird as such and be done. At first glance, this makes sense: if it looks like an orchard oriole, then it probably is.

But there is more to the story. My brother pointed out that Baltimore orioles are much more common in this part of the U.S., and some Baltimore orioles are dull-colored. In my neighborhood, you would expect to find more black-and-dull-orange Baltimore orioles than black-and-dull-orange orchard orioles. So, according to my brother, the bird was more likely to be a Baltimore oriole.*

This is a totally different conclusion!

Bayes’s Theorem is a more general, formal version of this type of thinking. You use contextual information, i.e. previously known probabilities, to evaluate the probability of something you’re interested in.

Formally, it’s the statement that

P(A | B) = [P(B | A) P(A)] / P( B )

where A | B means “A given B” and P(…) is “Probability of …”.

I’ll explain that formula, and the mistake of Mlodinow’s doctor, in the upcoming posts.

References

[1] Mlodinow, L. (2009). The Drunkard’s walk: How randomness rules our lives.

*In the end, my brother concluded it was an orchard oriole after all. But it’s still a good example of this type of reasoning!