Conditional Probability and Bayes Theorem

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:

$$\begin{gathered} P(A|B)=\frac{P(A\cup B)}{P(B)} \\ P(B|A)=\frac{P(A\cup B)}{P(A)} \end{gathered}$$

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.