The Binomial Distribution

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

State the distribution, its parameters, and the probability event so your reasoning is clear. A valid formula or clearly specified calculator method can communicate the calculation; the scoring requirements depend on the question. A bare number does not show which probability you calculated.

Q: When can I use binomial for sampling without replacement?

For a random sample without replacement, a sample size no more than 10% of the population lets us treat trials as approximately independent for a binomial approximation. The exact trials are still dependent.

AP Statistics· AP Statistics CED, Effective Fall 2026: Unit 2, Topic 2.10 — The Binomial Distribution· 12 min read

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

Use a binomial model only when the four BINS conditions hold, or when a stated approximation is justified. When asked to justify the model, explain each relevant condition using the context.

📘 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 takes 10 free throws. Assume that the shots are independent and that the probability of making each shot is constant at 0.75. Under these assumptions, 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: Independence of the shots is given as a modeling assumption.
3
Check N: Fixed number of trials n=10 → satisfied.
4
Check S: The constant success probability p=0.75 is given as a modeling assumption.
5
Conclusion: Under the stated assumptions, the number of made shots follows B(10, 0.75). A historical shooting percentage alone would not establish independence or constant probability.

✓ Quick check

Identify which of the following scenarios is binomial:

Drawing 6 cards without replacement from a standard 52-card deck and counting aces
- Valid binomial
- Not binomial, violates 10% condition
- Not binomial, no fixed n
Reveal answer
Not binomial, violates 10% condition —
The draws are dependent, and 6/52 ≈ 11.5% exceeds 10%, so the 10% condition does not justify a binomial approximation. The number of draws is fixed at 6.

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

Simulation (CED 2.10.D.1): to estimate P(X ≥ 7) for X ~ B(10, 0.75), let each independent uniformly generated random integer from 1 to 100 represent one shot, with 1–75 a success. Generate 10 integers per simulated game and count successes. Repeat many games; estimate the probability by the proportion with at least 7 successes. This simulated estimate varies between runs and can be compared with an exact binomial calculation.

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

For 0<p<1, a binomial distribution is symmetric when p=0.5, right-skewed when p<0.5, and left-skewed when p>0.5. The amount of skewness depends on both n and p; as n increases with p fixed, the distribution becomes more nearly 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 p=0.2<0.5, the distribution is right-skewed. Since np=4<10, the Large Counts condition fails, so use exact binomial probabilities rather than the usual normal approximation.

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:

For random sampling without replacement, confirm the sample size is no more than 10% of the population before treating trials as approximately independent.

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:

Failing the Large Counts condition means the usual normal approximation is not justified; it does not necessarily mean that the distribution is strongly skewed.

Correct move:

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

Wrong move:

Reporting a probability without identifying the distribution and event

Why:

A bare calculator output does not explain which probability was found. A full combination expansion is not a universal scoring requirement.

Correct move:

State the binomial parameters and event, then show a valid formula or clearly specified calculator calculation as the question requires.

6. Quick Reference Cheatsheet

Component	Formula / Rule	Communication / Check
Binomial Conditions	BINS: Binary, Independent, Fixed n, Constant p	Justify the relevant conditions in context when asked
Exact Probability	$P (X = k) = (k n) p^{k} (1 - p)^{n - k}$	Identify the distribution, parameters, and event
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	$n p \geq 10, n (1 - p) \geq 10$	Connection to Topic 3.2: check both counts

7. Frequently Asked

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

State the distribution, its parameters, and the probability event so your reasoning is clear. A valid formula or clearly specified calculator method can communicate the calculation; the scoring requirements depend on the question. A bare number does not show which probability you calculated.

When can I use binomial for sampling without replacement?

For a random sample without replacement, a sample size no more than 10% of the population lets us treat trials as approximately independent for a binomial approximation. The exact trials are still dependent.

What's Next

Binomial models lead to the sampling distribution of a sample proportion and inference for population proportions. Next, connect success counts to sample proportions and examine how repeated random samples produce sampling variability.