Maths in a minute: Bayes' theorem for beginners

Rational decisions should be based on evidence. This means keeping an open mind: when new evidence comes in, you should update your thinking.

Luckily mathematics has a tool to help you with this. It's called Bayes' theorem and belongs to the realm of probability theory.

To see how Bayes' theorem works, consider the following scenario. It's Sunday morning and you're still in bed. You haven't opened the blinds yet, but you know it's meant to be a nice day. Only a 10% chance of rain.

Then you hear the cat flap go. Shortly after, the cat walks into the bedroom soaking wet. That makes it a lot more likely that it's actually raining. But how likely?

Let's write $P(R)$ for the prior probability of rain. That's the estimate of the probability of rain before the new evidence involving the wet cat.  We write $P(W)$ for the prior probability that the cat is wet, whether it's raining or not. And we write $P(W|R)$ for the probability the cat is wet given that it's raining.

Finally, we write $P(R|W)$ for the probability that it's raining given that the cat is wet. That is the updated probability we are looking for in the light of the new evidence.

Bayes' theorem tells us that

$$P(R|W) = \frac{P(W|R)P(R)}{P(W)}.$$

The idea is that the probabilities on the right hand side of this equation are easier to estimate, so we can work out the left hand side. Let's have a go.

The prior chance of rain was 10%, so 

$$P(R)=0.1$$

The probability that the cat is wet given that it's raining, $P(W|R)$, is quite high since the cat would have found it hard to hide from the rain when crossing the garden to enter the cat flap. Let's put this chance at 90% so

$$P(W|R)=0.9.$$

As for $P(W)$, the cat absolutely hates water. There's only a small chance he will get wet for reasons other than rain. So the overall chance of a wet cat is just a little bit higher than the chance of rain. Let's put it at 11% . So 

$$P(W)=0.11.$$

We now have all the numbers to work out that

$$P(R|W) = \frac{P(W|R)P(R)}{P(W)} = \frac{0.9 \times 0.1}{0.11} = 0.82.$$

The updated probability that it is raining in the light of the new evidence is therefore 0.82, which translates to 82%.

Bayes' theorem is useful in a huge range of contexts and a staple in scientific reasoning. To find out how Bayes' theorem can help you understand the outcome of medical tests, see this article. And to learn about it in more detail, including some other interesting applications, see this article.