Bayes Thereom – broken down


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(Al) × P(B | Al)

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