Derivation of Bayes' Theorem

Bayes' Theorem is derived from the definition of conditional probability.

Definition of Conditional Probability:

The probability of given is:

Similarly, the probability of given is:

Where means intersection, or joint probability, or the probability that and occur. Rearrange both equations:

From :

From :

Equate the two expressions for and since both equations describe , set them equal:

Then we solve for , divide through by (assuming ) to get the final form of Bayes' Theorem:

where