# Bayesian probability – Relevance to poker

Bayesian probability involves using new or additional information to update prior probabilities. At poker.betzoo.uk 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.