Formula Sheet

Probability Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

8 formulas · 8 subtopics

P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) \neq 0

💡 Read as 'probability of A given B'. Reduces the sample space to B.

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)}

💡 Use when you know the result and want to find which cause produced it. 'Reverse' conditional probability.

P(A) = \sum_{i=1}^{n} P(E_i) \cdot P(A|E_i)

💡 E₁, E₂, ..., Eₙ must be a partition of the sample space (mutually exclusive, exhaustive).

P(X = r) = {}^nC_r p^r q^{n-r}, \quad q = 1-p

💡 n independent trials, each with success probability p. Mean = np, Variance = npq.

⚠ Forgetting that trials must be independent with constant probability for binomial to apply.

E(X) = np, \quad \text{Var}(X) = npq, \quad \sigma = \sqrt{npq}

💡 For binomial, Var(X) < E(X) since q < 1. If Var > Mean, it's NOT binomial.

P(A \cap B) = P(A) \cdot P(B) \iff A, B \text{ independent}

💡 Independence ≠ mutually exclusive. If A and B are mutually exclusive and both have non-zero probability, they are NOT independent.

E(X) = \sum x_i P(x_i), \quad \text{Var}(X) = E(X^2) - [E(X)]^2

💡 Var(aX+b) = a²Var(X). E(aX+b) = aE(X)+b. Variance is always non-negative.

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

💡 For mutually exclusive events: P(A ∪ B) = P(A) + P(B). Extend for 3 events using inclusion-exclusion.