Binomial distribution

IB Mathematics AI SL· Unit 4: Statistics and Probability, Topic 9· 25 min read

1. Conditions for a Binomial Distribution★★☆☆☆⏱ 6 min

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.

📘 Definition

Binomial Distribution

X \sim B(n, p)

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

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

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

2. Calculating Binomial Probabilities★★★☆☆⏱ 8 min

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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.

3. Expected Value and Variance★★☆☆☆⏱ 5 min

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: $σ = V a r (X)$ .

📐 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: $E (X) = n p = 100 \times 0.5 = 50$
3
Calculate variance: $V a r (X) = n p (1 - p) = 100 \times 0.5 \times 0.5 = 25$
4
Calculate standard deviation: $σ = 25 = 5$

4. Real-World Problem Solving★★★★☆⏱ 6 min

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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 $E (X) = n p = 20 \times 0.3 = 6$

✓ Quick check

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

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
Reveal answer
B: Number of red cards drawing 5 cards without replacement from a deck —
Correct! Drawing without replacement means trials are not independent, and probability changes after each draw, so the binomial conditions are not satisfied.

5. Common Pitfalls

Wrong move:

Using binomial distribution for sampling without replacement from a small population

Why:

Trials are not independent, so the constant probability condition fails

Correct move:

Only use binomial for sampling with replacement, or large populations where the change in probability is negligible

Wrong move:

Calculating $P (X \geq k)$ as $1 - P (X \leq k)$

Why:

This incorrectly excludes $k$ from the result, giving a value that is too low

Correct move:

Always use $P (X \geq k) = 1 - P (X \leq k - 1)$ and double check the inequality direction

Wrong move:

Rounding $p$ or intermediate probabilities early

Why:

Early rounding leads to loss of accuracy in the final result, costing marks

Correct move:

Keep all decimals in your GDC until the final step, only round the final answer to the required precision

Wrong move:

Forgetting to check binomial conditions before using the model

Why:

Exam questions often include scenarios where conditions are not met, and you lose marks for assuming binomial

Correct move:

Always explicitly check the four conditions before stating that the binomial distribution is appropriate

6. Quick Reference 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)
$P (X = k)$	$(k n) p^{k} (1 - p)^{n - k}$	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	$E (X) = n p$	Can be a decimal
Variance	$V a r (X) = n p (1 - p)$	SD = $V a r (X)$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Binomial probability calculation
2021 · 2
Real-world binomial problem
2023 · 1
Expected value of binomial

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.