Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A binomial random variable X counts the number of successes in n trials.
- The four conditions: each trial has two outcomes; the number of trials n is fixed; the trials are independent; the probability of success p is the same on every trial.
- Write X ~ B(n, p). X takes the values 0, 1, 2, …, n.
- Probabilities can be calculated with the formula or technology, or estimated with a simulation.
Key relationships
| What you want | How to find it |
|---|---|
| P(X = x) | ₙCₓ · pˣ · (1 − p)ⁿ⁻ˣ, where ₙCₓ = n! ÷ [x!(n − x)!]; or binomial pdf |
| P(X ≤ x) | add P(0) to P(x); or binomial cdf |
| P(X ≥ x) | 1 − P(X ≤ x − 1) |
| P(X ≥ 1) | 1 − P(X = 0) = 1 − (1 − p)ⁿ |
| P(a ≤ X ≤ b) | P(X ≤ b) − P(X ≤ a − 1) |
| Mean | μ = np |
| Standard deviation | σ = √[ np(1 − p) ] (variance = np(1 − p)) |
| Simulation estimate | (number of simulated repetitions with the event) ÷ (total repetitions) |
Assumptions and conventions
- Justify “binomial” by stating each condition in context; to reject it, name the condition that fails.
- Independence is often an assumption: say why it is reasonable (separate pots, random selection from a very large population).
- Word to inequality: “at least 3” is X ≥ 3; “more than 3” is X ≥ 4; “at most 3” is X ≤ 3; “fewer than 3” is X ≤ 2.
- Round probabilities to 4 decimal places, but add unrounded values.
Mistakes to avoid
- Leaving out ₙCₓ, so finding the probability of one order only.
- Swapping the powers of p and 1 − p.
- Wrong boundary for “at least”, “more than” or “fewer than”.
- Reporting the variance np(1 − p) as σ.
- Calling a variable binomial when n is not fixed or draws are without replacement from a small group.
- Calculator syntax only: always name the distribution, n, p and the values.
- No interpretation: say what the probability, mean or σ means for the people or objects in the question.
Quick self-check
- X ~ B(4, 0.5). Find P(X = 2). (₄C₂ (0.5)²(0.5)² = 6 × 0.0625 = 0.375)
- X ~ B(50, 0.2). Find μ and σ. (μ = 10; σ = √8 ≈ 2.83)
- X ~ B(6, 0.1). Find P(X ≥ 1). (1 − 0.9⁶ = 0.4686)
- Draw 3 cards without replacement from a 10-card pack with 4 red cards; X = number of red cards. Binomial? (No: p changes after each draw, so the trials are not independent.)
Next: practice questions.