Bayesian probability involves using new or additional information to update prior probabilities. At UnfairPoker.com we think a basic understanding of how Bayesian probability works is vital for poker players, which is why we have written this article.

The probability of event A, given condition B is:

p(A|B) = p(B|A) x p(A) / p(B)

## Let’s look at a real world example:

Imagine there is a test for a disease that is 99% accurate – 99% of people who have the disease will be correctly identified as having the disease (1% will be incorrectly identified as not having the disease, when they actually have it). 99% of people who don’t have the disease will be correctly identified as not having the disease (1% will be incorrectly identified as having the disease when they don’t).

0.20% of the population have this disease.

You take this 99% accurate test, and receive a positive test. Should you be worried?

Imagine 100,000 people are tested.

- 99,800 don’t have the disease
- 98,802 will test negative
- 998 will test positive

- 200 have the disease
- 198 will test positive
- 2 will test negative

998 + 198 =1,196 will test positive

However, only 198 / 1,196 actually have the disease

This means even though you tested positive in this 99% accurate test you have only a 16.5% chance of having the disease.

## How does Bayesian probability apply to poker?

Let us imagine you observe a tell that you think is 99% accurate (we use accurate in the same way as above) in predicting whether an opponent has a strong hand in a particular situation. However, your range analysis tells you the opponent is likely to only have a strong hand 0.2% of the time. As in the example above, your opponent has a strong hand 16.5% of the time, despite your 99% accurate tell.

Is a new player to the table playing a value heavy style, given that you have observed him fold his entire first orbit in a 9 handed table? You will find that you cannot necessarily conclude player A is not necessarily value heavy. However, it is more likely, than if you didn’t have this information.

There are infinite other ways Bayesian probability will help you in your understanding of poker.