Conditional Probability

See also: Conditional Distribution

def. Conditional Probability. The probability of event A given that event B has occurred is:

$$P(A|B):=\frac{P(A\cap B)}{P(B)}$$

• If $\Omega$’s elements are all equally likely: $P(A|B)=\frac{\#(A\cap B)}{\#B}$
• Multiplication Rule: $$\begin{gathered} P(A\cap B)=P(A|B)\cdot P(B) \\ =P(B|A)\cdot P(A) \end{gathered}$$
• Compliment Rule: $P(A|B)=1-P(A^C|B)$

thm. Chained Conditional Probability. For three events $A,B,C$:

$$\begin{gathered} P(ABC)=P(A)\cdot \frac{P(AB)}{P(A)}\cdot \frac{P(ABC)}{P(AB)} \\ =P(A)\cdot P(B|A)\cdot P(C|AB) \end{gathered}$$

def. Bayes’ Theorem. For events A, B:

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

Visualization.

…for Stochastic Calculus §

let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space. We define: def. Conditional Probability against Events of $A,B\in \Omega$ as $\mathbb{P}(A|B):=\mathbb{E}[1_A|B]$, the conditional expected value, which is defined as an integral/sum ${\int }_B{1}_{A}d\mathbb{P}$ def. Conditional Probability against