Probability and Statistical Distributions

Probability

• P(event) is between 0 and 1; all outcomes sum to 1; P(A′) = 1 − P(A).
• Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Mutually exclusive events can't both happen: P(A ∩ B) = 0.
• Independent events: P(A ∩ B) = P(A) × P(B).
• Conditional probability: P(A|B) = P(A ∩ B) ÷ P(B).
• Venn diagrams and tree diagrams organise these calculations.

Random variables and distributions

A discrete random variable X takes specific values, each with a probability. The probabilities in its distribution must sum to 1.

The binomial distribution

Used when there are n independent trials, each with two outcomes and a constant probability p of success. Written X ~ B(n, p).

P(X = r) = nCr · pʳ · (1 − p)^(n−r)

• Mean = np; variance = np(1 − p).
• Conditions: fixed n, independent trials, two outcomes, constant p.

Cumulative probabilities

P(X ≤ r) sums probabilities up to r — often read from tables or a calculator. Use P(X ≥ r) = 1 − P(X ≤ r−1).

Sampling

• A population is everything of interest; a sample is a subset used to estimate population features.
• Random sampling (each member equally likely) reduces bias; other methods include systematic, stratified and opportunity sampling.
• Larger samples generally give more reliable estimates.

Worked example

A biased coin has P(heads) = 0.3. It is flipped 5 times. Find P(exactly 2 heads).

• X ~ B(5, 0.3). P(X = 2) = 5C2 × 0.3² × 0.7³ = 10 × 0.09 × 0.343 = 0.309 (3 s.f.). ✓

Common mistakes

• Multiplying probabilities for events that aren't independent.
• Forgetting the (1 − p)^(n−r) factor in the binomial formula.
• Confusing P(X ≥ r) with P(X > r) when using cumulative tables.

Exam tips

• Check the binomial conditions before using B(n, p).
• Use P(A ∩ B) = P(A)P(B) only for independent events.
• For "at least" problems, use the complement (1 − P(fewer)).

Key facts to remember

• Probability rules: P(A∪B) = P(A) + P(B) − P(A∩B); independent ⇒ P(A∩B) = P(A)P(B); P(A|B) = P(A∩B)/P(B).
• Binomial X ~ B(n, p): P(X = r) = nCr pʳ(1−p)^(n−r); mean = np; needs fixed n, independent trials, constant p.
• Use random sampling to reduce bias; larger samples are more reliable.