What is Bayes’ Theorem?

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.

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


  • 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.


  1. Math is Fun
  2. brilliant.org
  3. wikipedia.org

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