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Probability Formulas

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Expected Value

$E(X) = \sum x_i P(x_i)$

Binomial Probability Formula

$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$

Variance of a Probability Distribution

$\sigma^2(X) = E[(X - E(X))^2] = \sum (x_i - E(X))^2 P(x_i)$

Probability of the Complement

$P(A') = 1 - P(A)$

Conditional Probability

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

Multiplication Rule of Probability (Intersection)

$P(A \cap B) = P(A) \times P(B|A)$

Addition Rule of Probability (Union)

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Bayes' Theorem

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

Law of Total Probability

$P(A) = \sum_{i=1}^n P(A|B_i) \times P(B_i)$

Standard Deviation of a Probability Distribution

$\sigma(X) = \sqrt{\sigma^2(X)}$

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