Formulas

Basic Probability

• \(P(A) = \frac{\#A}{\#\Omega}\) (for equally likely outcomes)
• \(P(A^c) = 1 - P(A)\) (complement rule)
• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) (union of two events)
• \(0 \leq P(A) \leq 1\) (probability bounds)

Counting Methods

• \(P_{n,k} = \frac{n!}{(n-k)!}\) (permutation, order matters)
• \(C_{n,k} = \frac{n!}{k!(n-k)!}\) (combination, order doesn't matter)

Conditional Probability and Bayes' Theorem

• \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) (conditional probability)
• \(P(A \cap B) = P(A|B) \cdot P(B)\) (multiplication rule)
• \(P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}\) (Bayes' theorem)
• \(P(A) = \sum_i P(A|B_i)P(B_i)\) (law of total probability)

Expectation and Variance

• \(E[X] = \sum x \cdot P(X=x)\) (expectation, discrete)
• \(E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx\) (expectation, continuous)
• \(\text{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2\) (variance)
• \(E[X^2] = \sum x^2 \cdot P(X=x)\) (second moment, discrete)
• \(E[X^2] = \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx\) (second moment, continuous)

Binomial Distribution

• \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) (binomial probability mass function)
• \(E[X] = np\) (binomial mean)
• \(\text{Var}(X) = np(1-p)\) (binomial variance)

Bernoulli Distribution

• \(\text{Var}(X_i) = p(1-p)\) (Bernoulli trial variance)

Independence and Multivariate Distributions

• \(P(X=x, Y=y) = P(X=x) \cdot P(Y=y)\) (independence condition for all \(x, y\))
• \(\text{Cov}(X,Y) = E[XY] - E[X]E[Y]\) (covariance)
• \(\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)\) (variance of sum)
• \(\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y)\) (for independent variables)
• \(E[XY] = \sum_x \sum_y xy \cdot P(X=x, Y=y)\) (joint expectation, discrete)

Marginal Distributions

• \(P(X=x) = \sum_y P(X=x, Y=y)\) (marginal probability, discrete)
• \(P(Y=y) = \sum_x P(X=x, Y=y)\) (marginal probability, discrete)

Continuous Distributions

• \(P(a \leq X \leq b) = \int_a^b f(x) \, dx\) (probability over interval)
• \(\int_{-\infty}^{\infty} f(x) \, dx = 1\) (total probability for continuous)
• \(f_Y(y) = \frac{d}{dy}[F_Y(y)]\) (PDF from CDF)
• \(F_Y(y) = P(Y \leq y)\) (CDF definition)