Probability and distributions: study guide

Key ideas

Probability gives numbers between 0 and 1 to uncertain events. A few rules, for combining events, counting arrangements and standardizing a normal value, cover most problems.

• The probability of A or B is P(A) + P(B) − P(A and B). For independent events, P(A and B) = P(A) × P(B).
• Conditional probability is P(A | B) = P(A and B) ÷ P(B).
• Count ordered selections with permutations nPr and unordered ones with combinations nCr. The binomial probability of k successes in n trials is C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ.
• A z-score z = (x − μ) ÷ σ says how many standard deviations a value is from the mean. The expected value is Σ x·P(x).

Key equations

• z = (x − μ) ÷ σ
• P(k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• P(A | B) = P(A ∩ B) ÷ P(B)
• nPr = n! ÷ (n − r)!, nCr = n! ÷ (r!(n − r)!)

Worked example

An IQ of 115 with a mean of 100 and a standard deviation of 15 has z = (115 − 100) ÷ 15 = 1. The probability of exactly 2 heads in 5 fair tosses is C(5, 2) × 0.5⁵ = 10 ÷ 32 = 0.3125. Choosing and ordering 3 of 8 runners for the podium gives 8P3 = 336, while choosing a 3-person team gives 8C3 = 56.

Common mistakes

• Adding P(A) and P(B) without subtracting the overlap, which can give a probability above 1.
• Using combinations when the order matters, or permutations when it does not.
• Treating dependent events as independent and multiplying.

Practice

Test yourself with practice questions on this topic: Probability and distributions