public-notes/Conditional Probability and Bayes Theorem.md

#Math #Probability

Conditional Probability

Conditional probability, or the probability of A given B is:

P(A|B)

Let's start with the probability of P(A \cup B). We know that when P(A | B), B is given to be true. Therefore, we must divide the probably of P(A \cup B) by P(B).

P(A | B) = \frac {P(A \cup B)} {P(B)}

This defines P(A | B) for events. When P(A | B) = P(A), A and B are independent.

Bayes’ Theorem

Let's start with the definitions of conditional probability:

P(A | B) = \frac {P(A \cup B)} {P(B)} \\
P(B | A) = \frac {P(A \cup B)} {P(A)}

Rearrange the second equation to define P(A \cup B):

P(A \cup B) = P(A) P(B | A)

Now substitute that equation into the first equation:

P(A | B) = \frac {P(A) P(B | A)} {P(B)}

The above equation is Bayes' Theorem for events.