Probability, Random Variables, and Probability Distributions
Probability rules and conditional probability
Probabilities lie in [0, 1] and sum to 1 over a sample space. Use the complement rule P(Aᶜ) = 1 − P(A), the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and the multiplication rule for independent events P(A ∩ B) = P(A)P(B). Independence is a modeling assumption - verify it, don't assume it from a problem's wording. Conditional probability is P(A|B) = P(A ∩ B)/P(B). The most common trap is confusing P(A|B) with P(B|A); the tree-diagram method (multiply along branches, add across) handles multi-stage problems cleanly.
Discrete and binomial random variables
For a random variable X, the mean is μ = Σx·P(x) and variance is σ² = Σ(x − μ)²·P(x). A binomial setting requires: fixed n trials, two outcomes, constant probability p, and independent trials. Then P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with μ = np and σ = √(np(1−p)). The geometric distribution counts trials until the first success: P(X = k) = (1−p)ᵏ⁻¹p. Watch for "at least one" questions - use the complement.
Continuous distributions
The normal model is the workhorse; use z-scores and the table or calculator. A density curve's total area is 1, and probability equals area under the curve. Sampling distributions of a sample proportion have mean p and standard deviation √(p(1−p)/n), approximately normal when np and n(1−p) are both at least 10.
Rule of thumb. Before computing, ask: "Is this asking for a joint, marginal, or conditional probability?" Write the notation from the question, then choose the rule.