Summary

Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, for $P(B) > 0$. It redefines the sample space to $B$.

Multiplication Theorem: $P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$.

Independent Events: $A$ and $B$ are independent iff $P(A \cap B) = P(A) \cdot P(B)$. This implies $P(A|B) = P(A)$ and $P(B|A) = P(B)$.

Total Probability Theorem: If events $E_1, E_2, \dots, E_n$ form a partition of the sample space, then $P(A) = \sum_{i=1}^n P(E_i) \cdot P(A|E_i)$.

Bayes' Theorem: $P(E_i|A) = \frac{P(E_i) \cdot P(A|E_i)}{\sum_{j=1}^n P(E_j) \cdot P(A|E_j)}$. Used to find "reverse" probabilities.

Random Variable: A function $X: S \to \mathbb{R}$. Its probability distribution lists all values $x_i$ with $P(X = x_i)$.

Mean (Expectation): $\mu = E(X) = \sum x_i \cdot P(X = x_i)$.

Variance: $\text{Var}(X) = \sum (x_i - \mu)^2 \cdot P(X = x_i) = E(X^2) - [E(X)]^2$. …