To arrive at an understanding of Bayes Theorem, we begin with the

## Definition of a conditional probability

**P(A|B)** = translates to “The probability of A happening, given that event B has occurred”.

P(A|B) can be rewritten like

**P(A | B) = P(A ∩ B)**

** ————**

** P(B)**

Also, P(A ∩ B) = P(A | B) × P(B) (*i.e. the probability of A and B*).

And because P(A ∩ B) = can also be expressed as P(B ∩ A)

We can say P(B ∩ A) = P(B | A) × P(A)

**Therefore, we can also express “the probability of A happening given event B” as**

**P(A | B) = P(B | A) × P(A) **

** ——————–**

** P(B)**

We can further expand the probability of B by writing it as

P(B) = P(A) × P(B | A) + P(Al) × P(B | Al )

(in normal language, the probability of B is equal to the probability of A multiplied by the probability of A given B, plus the probability of the complement of A multiplied by the probability of the complement of A given B)

**Thus resulting in the Law of Total Probability. **

This law enables us to find the total probability of a particular event based on conditional probabilities. Also, our new expression can be substituted into our formula for “the probability of A happening given event B” as the denominator to give us Bayes Theorem.

Bayes Theorem provides a means of finding reverse conditional probabilities when you don’t know every probability up front.

## Bayes Theorem

**P(A | B) = P(A) × P(B | A) **

** ——————————-**

** P(A) × P(B | A) + P(****A****l****) × P(B | ****A****l****)**