# Binomial distribution

> IB Mathematics Analysis and Approaches Higher Level · IB Math AA HL
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u4-binomial-distribution/

This module covers the 4 defining conditions of a binomial setting, the binomial probability mass function, derived expectation and variance formulas, and standard IB exam problem workflows.

**Prerequisites:** [Discrete random variables basics](https://www.owlsprep.com/study/ib-math-aa-hl-u4-discrete-random-variables/); [Combinations (n choose k)](https://www.owlsprep.com/study/ib-math-aa-hl-u3-permutations-combinations/)

## Learning objectives

- Identify the four required conditions for a valid binomial setting
- Apply the binomial probability mass function to calculate exact and cumulative probabilities
- Derive and use closed-form formulas for the expectation and variance of a binomial distribution
- Solve contextual IB AA HL problems involving binomial distributions with GDC support

## Defining the Binomial Setting

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'.

**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.

> **Binomial Conditions Mnemonic**
>
> BINS: Binary outcome, Independent trials, Fixed Number of trials, constant Success probability

**Check your understanding**

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

1. Drawing 5 cards without replacement from a standard deck and counting aces drawn

   *Why:* Trials are not independent, as sampling without replacement changes the success probability each time.

2. Rolling a fair 6-sided die 10 times and counting how many times you roll a 6

   *Why:* All BINS conditions are satisfied: binary outcome, independent rolls, n=10, p=1/6 constant.

## Binomial Probability Mass Function

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. $$= 35 \times (0.5)^7 = 35 / 128 ≈ 0.2734$$

**Exam command terms**

IB exam questions use specific command terms for binomial probability calculations:

- **Find P(X = k)** — Calculate exact probability for k successes, no further steps required

- **Show that P(X > 3) < 0.5** — You must present full working for the sum of probabilities to justify the inequality

## Expectation and Variance of Binomial Distribution

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:** 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. $$= \sum_{i=1}^n p = np$$

*Conclusion:* 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. $$Var(X) = 500 \times 0.02 \times 0.98 = 9.8$$

## Cumulative Binomial Probability Calculations

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.

> **GDC Best Practice**
>
> Always confirm your GDC is set to the correct parameters for cumulative vs individual probability, as mixing these up is a very common mark deduction.

**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. $$P(X ≥9) = 1 - 0.7747 ≈ 0.225$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using the binomial distribution for sampling without replacement
  - Why it fails: Trials are not independent, so the BINS conditions are violated
  - Correct: Use the hypergeometric distribution instead for sampling without replacement scenarios
- **Wrong:** Calculating P(X ≥k) as 1 - P(X ≤k)
  - Why it fails: This excludes the term P(X=k), leading to an off-by-one error
  - Correct: Rewrite P(X ≥k) as 1 - P(X ≤ k-1) to include all valid terms
- **Wrong:** Forgetting to use combinations in the PMF, writing P(X=k) = p^k (1-p)^{n-k}
  - Why it fails: This only counts the probability of one specific ordered sequence of k successes, not all possible sequences
  - Correct: Multiply by the binomial coefficient n choose k to account for all permutations of successes and failures
- **Wrong:** Using the formula Var(X) = np for binomial variance
  - Why it fails: This gives the expected value, not the variance, losing the (1-p) factor
  - Correct: Use the full variance formula Var(X) = np(1-p)
- **Wrong:** Rounding intermediate probability values early in multi-step problems
  - Why it fails: This introduces cascading rounding errors that make your final answer outside the acceptable IB mark range
  - Correct: Keep full unrounded values in your GDC memory until you reach the final step of the calculation

## 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.

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