# Binomial distribution

> IB Mathematics AI SL · Statistics and Probability
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-binomial-distribution/

This module covers the conditions for using a binomial distribution, how to calculate exact and cumulative probabilities, find expected value and variance, and apply binomial models to real-world problems common in IB AI SL exams.

**Prerequisites:** [Discrete random variables](https://www.owlsprep.com/study/ib-math-ai-sl-u4-discrete-random-variables/); [Basic probability rules](https://www.owlsprep.com/study/ib-math-ai-sl-u3-probability-fundamentals/)

## Learning objectives

- Identify the four conditions required for a binomial distribution
- Calculate binomial probabilities (exact and cumulative) using formula and GDC
- Find the expected value, variance and standard deviation of a binomial distribution
- Solve real-world exam problems involving binomial models

## Conditions for a Binomial Distribution

A binomial distribution models the number of successes (outcomes of interest) in a series of identical trials. It only applies if four core conditions are satisfied.

**Binomial Distribution** — A discrete probability distribution for the number of successes in a fixed sequence of independent Bernoulli trials.

*Notation:* X \sim B(n, p)

*Example:* Number of heads in 10 coin flips

- Fixed number of trials ($n$)
- Each trial is independent of all other trials
- Only two possible outcomes per trial (success/failure)
- Constant probability of success ($p$) across all trials

> **warning**
>
> Most exam errors come from failing to check these conditions before using a binomial model. Always confirm independence and fixed $n$ first.

**Worked example:** 5% of light bulbs produced at a factory are defective. A random sample of 10 bulbs is selected. Can the number of defective bulbs be modelled by a binomial distribution?

1. Check each condition one by one:
2. 1. Fixed number of trials: $n=10$, so condition is met
3. 2. Two outcomes: defective (success) or not defective (failure), condition met
4. 3. Independence: Random sampling means one bulb being defective does not affect another, condition met
5. 4. Constant probability: $p=0.05$ for all bulbs, condition met
6. Conclusion: Yes, the number of defective bulbs can be modelled as $X \sim B(10, 0.05)$

## Calculating Binomial Probabilities

To find the probability of exactly $k$ successes, we use the binomial probability formula. For IB AI SL, you will usually use your GDC to calculate probabilities, but you must know the formula for 'show that' questions.

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

For cumulative probabilities (probability of at most/least $k$ successes), we use the following rules:

- At most $k$ successes: $P(X \leq k)$ (use GDC binomial cdf directly)
- Fewer than $k$ successes: $P(X < k) = P(X \leq k-1)$
- More than $k$ successes: $P(X > k) = 1 - P(X \leq k)$
- At least $k$ successes: $P(X \geq k) = 1 - P(X \leq k-1)$

**Worked example:** Given $X \sim B(10, 0.05)$, find $P(X \geq 1)$.

1. We know $P(X \geq 1) = 1 - P(X=0) = 1 - P(X \leq 0)$
2. Substitute into the formula for $P(X=0)$:
3. $$P(X=0) = \binom{10}{0} (0.05)^0 (0.95)^{10} = 1 \times 1 \times (0.95)^{10} \approx 0.5987$$
4. Calculate the final result: $P(X \geq 1) = 1 - 0.5987 = 0.4013$ (4 decimal places)
5. This matches the result from GDC binomial cdf, confirming the answer.

*Calculator:* allowed

## Expected Value and Variance

For any binomial distribution $X \sim B(n,p)$, we can calculate the mean (expected value) and variance using simple formulas, no need to construct the full probability distribution.

$$E(X) = np$$

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

The standard deviation is the square root of the variance: $\sigma = \sqrt{Var(X)}$.

> **info**
>
> The expected value can be a decimal, even though $X$ is discrete. Do not round it to an integer unless the question explicitly asks for it.

**Worked example:** A fair coin is flipped 100 times, $X$ is the number of heads. Find the expected value and standard deviation of $X$.

1. First confirm the distribution: $X \sim B(100, 0.5)$
2. Calculate expected value: $E(X) = np = 100 \times 0.5 = 50$
3. Calculate variance: $Var(X) = np(1-p) = 100 \times 0.5 \times 0.5 = 25$
4. Calculate standard deviation: $\sigma = \sqrt{25} = 5$

## Real-World Problem Solving

Almost all IB exam questions for this topic ask you to apply binomial distribution to a real context, following a consistent step-by-step process.

**Worked example:** 30% of adults in a population wear glasses. 20 adults are selected at random. Find the probability that more than 5 wear glasses, and the expected number of glasses wearers.

1. Step 1: Check conditions: fixed $n=20$, independent random sample, two outcomes, constant $p=0.3$. So $X \sim B(20, 0.3)$
2. Step 2: We need $P(X > 5) = 1 - P(X \leq 5)$
3. Step 3: Use GDC binomial cdf to get $P(X \leq 5) \approx 0.4164$
4. Step 4: Calculate $P(X > 5) = 1 - 0.4164 = 0.584$ (3 significant figures)
5. Step 5: Expected value $E(X) = np = 20 \times 0.3 = 6$

**Check your understanding**

Which of the following scenarios cannot be modelled by a binomial distribution?

1. Which cannot use a binomial model?

   - A: Number of sixes when rolling a die 10 times
   - B: Number of red cards drawing 5 cards without replacement from a deck
   - C: Number of voters who support Candidate A in a random sample of 100
   - D: Number of defective items in a batch of 20

   *Why:* Correct! Drawing without replacement means trials are not independent, and probability changes after each draw, so the binomial conditions are not satisfied.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using binomial distribution for sampling without replacement from a small population
  - Why it fails: Trials are not independent, so the constant probability condition fails
  - Correct: Only use binomial for sampling with replacement, or large populations where the change in probability is negligible
- **Wrong:** Calculating $P(X \geq k)$ as $1 - P(X \leq k)$
  - Why it fails: This incorrectly excludes $k$ from the result, giving a value that is too low
  - Correct: Always use $P(X \geq k) = 1 - P(X \leq k-1)$ and double check the inequality direction
- **Wrong:** Rounding $p$ or intermediate probabilities early
  - Why it fails: Early rounding leads to loss of accuracy in the final result, costing marks
  - Correct: Keep all decimals in your GDC until the final step, only round the final answer to the required precision
- **Wrong:** Forgetting to check binomial conditions before using the model
  - Why it fails: Exam questions often include scenarios where conditions are not met, and you lose marks for assuming binomial
  - Correct: Always explicitly check the four conditions before stating that the binomial distribution is appropriate

## Cheatsheet

| Property | Formula/Rule | Exam Note |
| --- | --- | --- |
| Conditions | Fixed $n$, independent, 2 outcomes, constant $p$ | Check first for all questions |
| Notation | $X \sim B(n, p)$ | $n$ = trials, $p$ = P(success) |
| $P(X=k)$ | $\binom{n}{k} p^k (1-p)^{n-k}$ | Use GDC binompdf |
| $P(X \leq k)$ | Cumulative sum of probabilities | Use GDC binomcdf |
| $P(X \geq k)$ | $1 - P(X \leq k-1)$ | Most common inequality mistake |
| Expected Value | $E(X) = np$ | Can be a decimal |
| Variance | $Var(X) = np(1-p)$ | SD = $\sqrt{Var(X)}$ |

## What's next

Binomial distribution is one of the most frequently tested topics in IB AI SL probability and statistics, and it forms the foundation for many higher-level concepts you will encounter. Mastering condition checking and cumulative probability calculation will help you earn full marks on both Paper 1 and Paper 2 questions. Binomial models are also the basis for hypothesis testing for proportions, a key topic in Unit 5 of the IB AI SL syllabus. The skills you learn here will transfer directly to that topic, and to any other probability problems involving discrete outcomes.

- [Normal distribution and applications](https://www.owlsprep.com/study/ib-math-ai-sl-u4-normal-distribution-and-applications/)
- [Calculus](https://www.owlsprep.com/study/ib-math-ai-sl-u5-overview/)
- [Derivative as gradient of tangent and rate of change](https://www.owlsprep.com/study/ib-math-ai-sl-u5-derivative-as-gradient-of-tangent/)

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