# What is Bayes’ Theorem?

Bayes’s theorem is a way of finding a probability when we have certain other probabilities. Bayes’s theorem is stated mathematically as the following equation:

**
**Where A and B are events and P(B) ≠ 0,

P(A | B) is also a conditional probability: the likelihood of event A occurring given B is true.

P(B | A) is also a conditional probability: the likelihood of event B occurring given A is true.

P(A) and P(B) are conditional probabilities of observing A and B respectively.

Lets take an example from mathisfun.com . You can check other examples from this site.

**Example:
**Let us say P(Fire) means how often there is fire, and P(Smoke) means how often we see smoke, then:

P (Fire | Smoke) means how often there is fire when we can see smoke

P(Smoke | Fire) means how often we can see smoke when there fire

Given,

- dangerous fires are rare (1%) i.e. P(Fire)
- but smoke is fairly common (10%) due to barbecues i.e. P(Smoke)
- and 90% of dangerous fires make smoke i.e. P(Smoke | Fire)

Probability of dangerous fire when there is smoke:

P( Fire | Smoke) = P(Fire) x P(Smoke | Fire) / P(Smoke)

= 1 % x 90% / 10%

= 9%

So, it is still worth checking out any smoke to be sure.

You can check other example from below link.

**References:**