The Binomial Distribution

AP Statistics· Unit 4: Probability, Random Variables, and Probability Distributions· 12 min read

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

A binomial distribution only applies if all four required conditions are met. AP exam questions almost always require you to explicitly verify these conditions to earn full points, rather than assuming a scenario is binomial.

📘 Definition

BINS Conditions

The four requirements for a binomial setting: Binary outcomes, Independent trials, Fixed number of trials, Constant probability of success.

📐 Worked Example

A basketball player makes 75% of their free throws, and takes 10 shots in a game. Is the number of made shots a binomial random variable?

1
Check B: Each shot has two outcomes (make or miss) → satisfied.
2
Check I: Each shot is independent of the others → satisfied.
3
Check N: Fixed number of trials n=10 → satisfied.
4
Check S: Probability of success p=0.75 for every shot → satisfied.
5
Conclusion: Yes, this is a valid binomial setting.

✓ Quick check

Identify which of the following scenarios is binomial:

Drawing 5 cards without replacement from a standard deck and counting aces, population size 52
- Valid binomial
- Not binomial, violates 10% condition
- Not binomial, no fixed n
Reveal answer
Not binomial, violates 10% condition —
Sample size 5 is ~9.6% of 52, just under 10% but trials are not strictly independent, so it is not a perfect binomial setting.

2. Binomial Probability Calculation★★★☆☆⏱ 3 min

The binomial probability formula calculates the exact probability of getting exactly k successes in n trials. It accounts for all different orderings of k successes and (n-k) failures across the fixed trials.

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

🔬 Derivation

Goal:

Derive the binomial probability formula

Starting from:

Each specific sequence of k successes and n-k failures has probability $p^{k} (1 - p)^{n - k}$

1
The number of unique sequences of k successes in n trials is given by the combination formula $(k n) = \frac{n !}{k ! ( n - k )!}$
2
Multiply the number of sequences by the probability of each individual sequence to get total probability

Result:

This produces the standard binomial probability formula.

📐 Worked Example

For the basketball player with n=10 shots and p=0.75, calculate the probability they make exactly 7 shots.

1
Identify parameters: n=10, k=7, p=0.75
2
$\binom{10}{7} = 120$
3
$P(X=7) = 120 * (0.75)^7 * (0.25)^3$
4
Calculate final value: P(X=7) ≈ 0.2503

3. Mean and Standard Deviation of Binomial Distributions★★☆☆☆⏱ 2 min

Binomial random variables have simple closed-form formulas for their expected value (mean) and standard deviation, no need to sum over all possible outcomes manually.

\mu_X = E(X) = np

\sigma_X = \sqrt{np(1-p)}

📐 Worked Example

Find and interpret the mean and standard deviation of the number of made free throws for the player taking 10 shots with p=0.75.

1
$\mu_X = 10 * 0.75 = 7.5$
2
Interpretation: Over many sets of 10 free throws, the player will make an average of 7.5 shots.
3
$\sigma_X = \sqrt{10 * 0.75 * 0.25} ≈ 1.369$
4
Interpretation: The number of made shots typically varies from 7.5 by about 1.37 shots.

4. Shape of the Binomial Distribution★★★☆☆⏱ 2 min

The shape of a binomial distribution depends on values of n and p. When p is close to 0 or 1, the distribution is skewed. When np ≥ 10 and n(1-p) ≥10, the distribution is approximately symmetric.

📐 Worked Example

Describe the shape of a binomial distribution with n=20, p=0.2

1
$np = 20 * 0.2 = 4 < 10$
2
$n(1-p) = 20 * 0.8 = 16 ≥10$
3
Since np < 10, the distribution is strongly right-skewed, not approximately normal.

5. Common Pitfalls

Wrong move:

Applying binomial to sampling without replacement and ignoring the 10% condition

Why:

Trials are not strictly independent, which violates a core BINS condition

Correct move:

Confirm the sample size is ≤10% of the population before using binomial approximation

Wrong move:

Calculating P(X ≥ k) as P(X > k) by excluding the k value

Why:

Binomial counts discrete successes, so ≥ includes the boundary value k

Correct move:

Rewrite P(X ≥k) as 1 - P(X ≤ k-1) to avoid off-by-one errors

Wrong move:

Stating the binomial mean as a probability in context

Why:

The expected value is a long-run average, not a guaranteed outcome for a single trial

Correct move:

Interpret the mean as the average result across many identical repetitions of the scenario

Wrong move:

Using the normal approximation for binomial when np < 10

Why:

The distribution is heavily skewed, and normal approximation will produce inaccurate probability values

Correct move:

Use exact binomial probability calculations when the Large Counts condition is not met

Wrong move:

Forgetting to show the combination term in the binomial formula on free response

Why:

AP graders will assume you used a calculator without understanding the underlying logic

Correct move:

Write out the full formula with the combination term before substituting values

6. Quick Reference Cheatsheet

Component	Formula / Rule	AP Exam Requirement
Binomial Conditions	BINS: Binary, Independent, Fixed n, Constant p	Explicitly verify all 4 for full credit
Exact Probability	$P (X = k) = (k n) p^{k} (1 - p)^{n - k}$	Show setup before calculator output
Mean	$μ_{X} = n p$	Interpret as long-run average
Standard Deviation	$σ_{X} = n p (1 - p)$	No square root simplification required
Normal Approximation	$n p \geq 10, n (1 - p) \geq 10$	Must state Large Counts condition first

7. Frequently Asked

Can I directly write the result from binompdf on my AP exam free response?

No. You must explicitly state the distribution parameters, write out the probability setup, and show your work before stating the final value to earn full credit. Only writing the calculator output will lose communication points.

When can I use binomial for sampling without replacement?

If your sample size is no more than 10% of the total population, the 10% condition is satisfied, and you can treat the trials as approximately independent to use the binomial distribution.

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.

2023 · Paper 1
Binomial probability FRQ calculation
2021 · Paper 2
Binomial condition MCQ set
2019 · Paper 1
Binomial mean interpretation FRQ

What's Next

Mastering the binomial distribution is a critical foundation for the rest of AP Statistics, as it directly underpins sampling distributions for sample proportions and inference for categorical data. You will use binomial logic to build intuition for the Central Limit Theorem, design hypothesis tests for population proportions, and calculate confidence intervals for survey results. This concept is also a frequent anchor for multi-part free response questions that combine probability, random variables, and inference. Next, you will explore the geometric distribution, the other core discrete distribution for counting trials until the first success, then move to sampling distributions that describe the behavior of sample statistics across repeated sampling.