Binomial distribution

IB Mathematics Analysis and Approaches Higher Level· Topic 4: Statistics and Probability, 4.11· 12 min read

1. Defining the Binomial Setting★★☆☆☆⏱ 3 min

A binomial distribution describes the number of successes in a fixed number of repeated, independent Bernoulli trials, where each trial has exactly two mutually exclusive outcomes labelled 'success' and 'failure'.

📘 Definition

Binomial Setting

A probabilistic scenario that satisfies all four conditions: fixed number of trials n, independent trials, constant success probability p, exactly two outcomes per trial.

✓ Quick check

Identify which of the following scenarios qualify as a valid binomial setting

Drawing 5 cards without replacement from a standard deck and counting aces drawn
Reveal answer
No —
Trials are not independent, as sampling without replacement changes the success probability each time.
Rolling a fair 6-sided die 10 times and counting how many times you roll a 6
Reveal answer
Yes —
All BINS conditions are satisfied: binary outcome, independent rolls, n=10, p=1/6 constant.

2. Binomial Probability Mass Function★★★☆☆⏱ 4 min

For a binomial random variable X ~ B(n,p), the probability of observing exactly k successes is calculated using the PMF that combines combinations and independent probability rules.

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0,1,2,...,n

📐 Worked Example

A fair coin is tossed 7 times. Calculate the probability of obtaining exactly 4 heads.

1
Define the distribution: X = number of heads in 7 tosses, so X ~ B(7, 0.5)
2
Substitute n=7, k=4, p=0.5 into the PMF
3
$P(X=4) = \binom{7}{4} (0.5)^4 (0.5)^{3}$
4
$= 35 \times (0.5)^7 = 35 / 128 ≈ 0.2734$

3. Expectation and Variance of Binomial Distribution★★★☆☆⏱ 3 min

Instead of summing over all possible values of k, you can use pre-derived closed-form formulas for the expected value and variance of a binomial distribution, which are valid for all n ≥ 1 and 0 ≤ p ≤1.

\mathbb{E}(X) = np, \quad \text{Var}(X) = np(1-p)

🔬 Derivation

Goal:

Derive E(X) for binomial distribution

Starting from:

X can be written as the sum of n independent Bernoulli random variables X_i, each with E(X_i) = p

1
Linearity of expectation gives E(X) = E(X_1 + X_2 + ... + X_n)
2
This expands to sum of individual expectations: E(X) = sum_{i=1}^n E(X_i)
3
$= \sum_{i=1}^n p = np$

Result:

The expectation of a binomial distribution is simply the product of the number of trials and the success probability.

📐 Worked Example

A factory produces defective components with probability 0.02 per item. For a batch of 500 components, find the expected number of defective items and the variance.

1
Define X ~ B(500, 0.02)
2
$E(X) = 500 \times 0.02 = 10$
3
$Var(X) = 500 \times 0.02 \times 0.98 = 9.8$

4. Cumulative Binomial Probability Calculations★★★★☆⏱ 4 min

✓ Calculator OK

Most IB exam problems require calculating cumulative probabilities such as P(X ≤ k) or P(X ≥ k), which can be computed directly using your GDC's built-in binomial CDF function to avoid summing many individual PMF terms.

📐 Worked Example

X ~ B(12, 0.6). Find P(X ≥ 9).

1
Rewrite the inequality to match standard CDF input: P(X ≥9) = 1 - P(X ≤8)
2
Use GDC binomcdf(n=12, p=0.6, k=8) to get P(X ≤8) ≈ 0.7747
3
$P(X ≥9) = 1 - 0.7747 ≈ 0.225$

5. Common Pitfalls

Wrong move:

Using the binomial distribution for sampling without replacement

Why:

Trials are not independent, so the BINS conditions are violated

Correct move:

Use the hypergeometric distribution instead for sampling without replacement scenarios

Wrong move:

Calculating P(X ≥k) as 1 - P(X ≤k)

Why:

This excludes the term P(X=k), leading to an off-by-one error

Correct move:

Rewrite P(X ≥k) as 1 - P(X ≤ k-1) to include all valid terms

Wrong move:

Forgetting to use combinations in the PMF, writing P(X=k) = p^k (1-p)^{n-k}

Why:

This only counts the probability of one specific ordered sequence of k successes, not all possible sequences

Correct move:

Multiply by the binomial coefficient n choose k to account for all permutations of successes and failures

Wrong move:

Using the formula Var(X) = np for binomial variance

Why:

This gives the expected value, not the variance, losing the (1-p) factor

Correct move:

Use the full variance formula Var(X) = np(1-p)

Wrong move:

Rounding intermediate probability values early in multi-step problems

Why:

This introduces cascading rounding errors that make your final answer outside the acceptable IB mark range

Correct move:

Keep full unrounded values in your GDC memory until you reach the final step of the calculation

6. Quick Reference Cheatsheet

Quantity	Formula for X ~ B(n,p)
PMF for exactly k successes	P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}
Cumulative probability up to k	P(X ≤k) = \sum_{i=0}^k \binom{n}{i} p^i (1-p)^{n-i}
Expected value	E(X) = np
Variance	Var(X) = np(1-p)

What's Next

Mastering binomial distributions is a critical foundation for later probability and statistics topics in AA HL, including modelling with the normal distribution and reasoning with conditional probability. You will frequently combine binomial probability rules with combinatorics and conditional probability to solve extended Paper 3 problem sets that carry 15-20% of your total statistics marks. Ensure you can quickly verify if a scenario meets the binomial conditions before selecting this model in exam contexts.