The Binomial Distribution

Mathematics· 9709 Paper 5, Section 5.4· 18 min read

1. Binomial Model Conditions and Notation★★☆☆☆⏱ 4 min

📘 Definition

Binomial Distribution

X ~ B(n, p)

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

Example:

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

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

📐 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.

2. Single Term Binomial Probability Formula★★★☆☆⏱ 5 min

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
$= 0.31104 \approx 0.311$

✓ Quick check

Test your understanding of single binomial probability calculations

If X ~ B(5, 0.5), what is P(X=3)?
- 0.3125
- 0.5
- 0.25
- 0.125
Reveal answer
0.3125 —
Use ⁵C₃ × 0.5³ × 0.5² = 10 × 0.125 × 0.25 = 0.3125

3. Cumulative Probability Boundary Rules★★★★☆⏱ 5 min

📐 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
$P(X ≥ 2) = 1 - (0.046656 + 0.186624) = 0.76672 \approx 0.767$

4. Expectation and Variance of Binomial Distribution★★★☆☆⏱ 4 min

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
$Var(X) = np(1-p) = 30 \times \frac{1}{6} \times \frac{5}{6} = \frac{25}{6} \approx 4.17$

5. Common Pitfalls

Wrong move:

Treating 'fewer than 5' as P(X ≤ 5)

Why:

Fewer than 5 means X can only be 0,1,2,3,4, so the upper bound is 4, not 5

Correct move:

Translate 'fewer than r' to P(X ≤ r-1) before performing any calculations

Wrong move:

Using the total population size as n when sampling without replacement

Why:

Sampling without replacement breaks the constant p condition, so the binomial model is invalid here

Correct move:

n is the fixed number of trials, not the total size of the population you are sampling from

Wrong move:

Rounding individual binomial terms to 3 s.f. before summing for cumulative probabilities

Why:

Intermediate rounding introduces large, avoidable errors in the final cumulative value

Correct move:

Keep full unrounded values for all intermediate steps, only round the final answer to 3 significant figures

Wrong move:

Defining 'success' as the less likely outcome by mistake

Why:

This swaps p and 1-p, leading to completely incorrect probability values

Correct move:

Explicitly write down what event counts as a success, and confirm its probability matches your value for p

Wrong move:

Calculating variance as np instead of np(1-p)

Why:

Confusing the expectation formula with the variance formula is a very common careless error

Correct move:

Cross-reference your working with the MF19 formula sheet to confirm you have selected the right expression

6. Quick Reference 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.