# The Binomial Distribution

> AP Statistics · AP Stats
> Source: https://www.owlsprep.com/study/ap-statistics-u11-the-binomial-distribution/

This module covers binomial setting conditions, probability calculations, summary statistics, and exam-aligned interpretation for AP Statistics assessment requirements.

**Prerequisites:** [Basic discrete random variable properties](https://www.owlsprep.com/study/ap-statistics-u10-discrete-random-variables/); [Core probability and combinatorics rules](https://www.owlsprep.com/study/ap-statistics-u10-basic-probability-rules/)

## Learning objectives

- Identify and verify the four required conditions for a valid binomial setting
- Calculate exact binomial probabilities using the formula and approved calculator functions
- Compute and interpret the mean and standard deviation of a binomial random variable
- Apply binomial distribution logic to answer AP exam free response and multiple-choice questions

## Verifying the Binomial Setting

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.

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

> **BINS Memory Hook**
>
> Use the BINS acronym to never miss a condition check on exam day: Binary, Independent, Number (fixed), Success (constant probability).

**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.

**Check your understanding**

Identify which of the following scenarios is binomial:

1. 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

   *Why:* 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.

## Binomial Probability Calculation

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:** 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 $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
2. Multiply the number of sequences by the probability of each individual sequence to get total probability

*Conclusion:* 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

**Exam command terms**

AP exam command terms for binomial probability questions have specific requirements:

- **Calculate** — Show full setup including n, p, and the formula before the final number *(Calculate P(X ≥ 2) requires writing 1 - P(X=0) - P(X=1) before computing the value.)*

## Mean and Standard Deviation of Binomial Distributions

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.

## Shape of the Binomial Distribution

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.

> **warning**
>
> You cannot use normal approximation for binomial distributions unless the Large Counts condition (np ≥ 10, n(1-p) ≥10) is explicitly satisfied.

**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.

## Common pitfalls

- **Wrong:** Applying binomial to sampling without replacement and ignoring the 10% condition
  - Why it fails: Trials are not strictly independent, which violates a core BINS condition
  - Correct: Confirm the sample size is ≤10% of the population before using binomial approximation
- **Wrong:** Calculating P(X ≥ k) as P(X > k) by excluding the k value
  - Why it fails: Binomial counts discrete successes, so ≥ includes the boundary value k
  - Correct: Rewrite P(X ≥k) as 1 - P(X ≤ k-1) to avoid off-by-one errors
- **Wrong:** Stating the binomial mean as a probability in context
  - Why it fails: The expected value is a long-run average, not a guaranteed outcome for a single trial
  - Correct: Interpret the mean as the average result across many identical repetitions of the scenario
- **Wrong:** Using the normal approximation for binomial when np < 10
  - Why it fails: The distribution is heavily skewed, and normal approximation will produce inaccurate probability values
  - Correct: Use exact binomial probability calculations when the Large Counts condition is not met
- **Wrong:** Forgetting to show the combination term in the binomial formula on free response
  - Why it fails: AP graders will assume you used a calculator without understanding the underlying logic
  - Correct: Write out the full formula with the combination term before substituting values

## 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 | $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ | Show setup before calculator output |
| Mean | $\mu_X = np$ | Interpret as long-run average |
| Standard Deviation | $\sigma_X = \sqrt{np(1-p)}$ | No square root simplification required |
| Normal Approximation | $np ≥ 10, n(1-p) ≥10$ | Must state Large Counts condition first |

## 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.

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