# The Binomial Distribution

> Mathematics · CIE A-Level 9709
> Source: https://www.owlsprep.com/study/cie-9709-u4-the-binomial-distribution/

This guide covers binomial model conditions, notation, probability formulae, cumulative boundary rules, and expectation/variance calculations for CIE 9709 Paper 5.

**Prerequisites:** [Basic permutations and combinations (ⁿCᵣ calculations)](https://www.owlsprep.com/study/cie-9709-u4-permutations-and-combinations/); [Discrete random variable fundamentals](https://www.owlsprep.com/study/cie-9709-u4-discrete-random-variables/)

## Learning objectives

- Identify valid binomial distribution scenarios and use X ~ B(n,p) standard notation
- Calculate individual and cumulative binomial probabilities using the official formula
- Compute expectation and variance for any binomial distribution
- Correctly interpret exam phrasing to avoid cumulative probability boundary errors

## Binomial Model Conditions and Notation

**Binomial Distribution** — A discrete probability model for the count of successes in n independent identical trials with constant success probability p.

*Notation:* X ~ B(n, p)

*Example:* Counting the number of heads in 10 fair coin tosses: X ~ B(10, 0.5)

1. There is a fixed, pre-defined number of n trials
2. Every trial has exactly two mutually exclusive outcomes (labelled success and failure)
3. All trials are fully independent of one another
4. The probability of success p is identical for every trial
5. The random variable X counts the number of successes in the n trials

> **warning**
>
> If any one of the four conditions fails, the binomial model is invalid. For example, sampling without replacement from a small population breaks the constant p rule.

**Worked example:** State whether a binomial distribution is suitable for this scenario: A bag holds 10 red and 10 blue marbles. 3 marbles are drawn without replacement, and X is the number of red marbles drawn. Justify your answer.

1. Check the four binomial conditions one by one:
2. Fixed n=3 trials, two outcomes (red/blue), but trials are not independent as p changes after each draw, so p is not constant.
3. Conclusion: Binomial distribution is NOT suitable.

## Single Term Binomial Probability Formula

$$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \quad \text{for } r = 0,1,2,...,n$$

This formula calculates the exact probability of getting exactly r successes out of n total trials. The term ⁿCᵣ counts the number of different ways to arrange r successes across n trials, pʳ is the probability of r consecutive successes, and (1-p)ⁿ⁻ʳ is the probability of the remaining n-r failures.

**Worked example:** Given X ~ B(6, 0.4), calculate P(X = 2), giving your answer to 3 significant figures.

1. Substitute n=6, r=2, p=0.4 into the standard formula:
2. $$P(X=2) = \binom{6}{2} (0.4)^2 (0.6)^{4}$$
3. $$= 15 \times 0.16 \times 0.1296$$
4. $$= 0.31104 \approx 0.311$$

**Check your understanding**

Test your understanding of single binomial probability calculations

1. If X ~ B(5, 0.5), what is P(X=3)?

   - 0.3125
   - 0.5
   - 0.25
   - 0.125

   *Why:* Use ⁵C₃ × 0.5³ × 0.5² = 10 × 0.125 × 0.25 = 0.3125

## Cumulative Probability Boundary Rules

**Exam command terms**

CIE exam questions use specific phrasing for cumulative binomial probabilities; misinterpreting boundaries is the top cause of lost marks on this topic.

- **at most r** — All values ≤ r, so P(X ≤ r)

- **fewer than r** — All values < r, so P(X ≤ r-1)

- **at least r** — All values ≥ r, so 1 - P(X ≤ r-1)

- **more than r** — All values > r, so 1 - P(X ≤ r)

**Worked example:** Given X ~ B(6, 0.4), calculate P(X ≥ 2), giving your answer to 3 significant figures.

1. Use the boundary rule: P(X ≥ 2) = 1 - P(X ≤ 1)
2. Expand P(X ≤1) = P(X=0) + P(X=1)
3. $$P(X=0) = \binom{6}{0} (0.4)^0 (0.6)^6 = 1 \times 1 \times 0.046656 = 0.046656$$
4. $$P(X=1) = \binom{6}{1} (0.4)^1 (0.6)^5 = 6 \times 0.4 \times 0.07776 = 0.186624$$
5. $$P(X ≥ 2) = 1 - (0.046656 + 0.186624) = 0.76672 \approx 0.767$$

> **tip**
>
> Always write down the full boundary translation step in your exam script, even if it feels obvious. Examiners award follow-through marks for correct steps even if you miscalculate a single term.

## Expectation and Variance of Binomial Distribution

$$E(X) = np$$

$$Var(X) = np(1-p) = npq$$

These two formulae are printed directly on the MF19 exam formula sheet, so you do not need to derive them in the exam. For large n this can be approximated by the normal distribution — see the Normal Distribution page.

**Worked example:** A fair 6-sided die is rolled 30 times. X is the number of times a 3 is rolled. Find E(X) and Var(X).

1. First identify n and p: n=30 trials, p = 1/6 (probability of rolling a 3)
2. $$E(X) = np = 30 \times \frac{1}{6} = 5$$
3. $$Var(X) = np(1-p) = 30 \times \frac{1}{6} \times \frac{5}{6} = \frac{25}{6} \approx 4.17$$

## Common pitfalls

- **Wrong:** Treating 'fewer than 5' as P(X ≤ 5)
  - Why it fails: Fewer than 5 means X can only be 0,1,2,3,4, so the upper bound is 4, not 5
  - Correct: Translate 'fewer than r' to P(X ≤ r-1) before performing any calculations
- **Wrong:** Using the total population size as n when sampling without replacement
  - Why it fails: Sampling without replacement breaks the constant p condition, so the binomial model is invalid here
  - Correct: n is the fixed number of trials, not the total size of the population you are sampling from
- **Wrong:** Rounding individual binomial terms to 3 s.f. before summing for cumulative probabilities
  - Why it fails: Intermediate rounding introduces large, avoidable errors in the final cumulative value
  - Correct: Keep full unrounded values for all intermediate steps, only round the final answer to 3 significant figures
- **Wrong:** Defining 'success' as the less likely outcome by mistake
  - Why it fails: This swaps p and 1-p, leading to completely incorrect probability values
  - Correct: Explicitly write down what event counts as a success, and confirm its probability matches your value for p
- **Wrong:** Calculating variance as np instead of np(1-p)
  - Why it fails: Confusing the expectation formula with the variance formula is a very common careless error
  - Correct: Cross-reference your working with the MF19 formula sheet to confirm you have selected the right expression

## Cheatsheet

| Description | Formula / Rule |
| --- | --- |
| Binomial distribution notation | X ~ B(n, p) |
| Single term probability | P(X=r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ |
| P(at most r successes) | P(X ≤ r) |
| P(at least r successes) | 1 - P(X ≤ r-1) |
| Expected value (mean) | E(X) = np |
| Variance | Var(X) = np(1-p) |

## What's next

Mastering the binomial distribution is a critical milestone for CIE 9709 Paper 5, as it forms the foundation for all other discrete probability models you will encounter. You will next extend your discrete random variable knowledge to the geometric distribution, the second named discrete model in the S1 syllabus, before moving on to continuous probability distributions including the normal distribution. The binomial expectation and variance rules you have learned here will also be reused when you work with sampling later in your course. Practice 3-4 past paper binomial questions to lock in the boundary translation skill, as this is the easiest way to secure full marks on this high-weight topic.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-9709-u4-the-binomial-distribution/