---
title: "AP Statistics 2.10: The Binomial Distribution"
description: "Review AP Statistics 2.10, including BINS conditions, binomial probabilities, and the mean and standard deviation of a binomial distribution."
canonical: "https://fiveable.me/ap-stats/unit-2/intro-binomial-distribution/study-guide/uvU3qsHmVgEuPjvhgaEC"
type: "study-guide"
subject: "AP Statistics"
unit: "Unit 2 – Probability, Random Variables, and Probability Distributions"
lastUpdated: "2026-06-30"
---

# AP Statistics 2.10: The Binomial Distribution

## Summary

Review AP Statistics 2.10, including BINS conditions, binomial probabilities, and the mean and standard deviation of a binomial distribution.

## Guide

A [binomial random variable](/ap-stats/key-terms/binomial-random-variable "fv-autolink") counts the number of successes in a fixed number of [independent trials](/ap-stats/key-terms/independent-trials "fv-autolink"), where each trial ends in success or failure and the probability of success $p$ stays the same. In this revised topic, you should be able to recognize a binomial setting, find probabilities, and use the shortcut formulas for the mean and standard deviation.

## Why This Matters for the AP Statistics Exam

The binomial distribution is one of the most common discrete models in [AP Statistics](/ap-stats "fv-autolink"). On the exam, you may need to recognize when a setting is binomial, compute the probability of a certain number of successes, or calculate the expected number of successes and the standard deviation of that count.

## Key Takeaways

- A binomial random variable counts successes, not trials.
- The **BINS** conditions are Binary [outcomes](/ap-stats/unit-2/estimating-probabilities-using-simulation/study-guide/ABwpnnUf4VCXeVbB9q72 "fv-autolink"), Independent trials, fixed Number of trials, and Same probability of success.
- Use $P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$ for the probability of exactly $x$ successes.
- For a binomial random variable, the mean is $\mu = np$ and the standard deviation is $\sigma = \sqrt{np(1-p)}$.

## What Makes a Setting Binomial

A **binomial random variable** counts the number of successes in a fixed number of independent trials. To recognize a binomial setting, check that all of these are true:

- there is a fixed number of trials, $n$
- each trial is independent
- each trial has only two outcomes: success or failure
- the probability of success $p$ is the same on every trial

Many AP Statistics teachers use the acronym **BINS**:

- **B**inary
- **I**ndependent
- fixed **N**umber of trials
- **S**ame probability of success

## Probabilities and Parameters

Once the conditions are met, you can answer two kinds of questions:

1. What is the probability of a certain number of successes?
2. What is the expected number of successes, and how much does that count typically vary?

### The Binomial Probability Formula

The probability that a binomial random variable $X$ has exactly $x$ successes in $n$ independent trials, with success probability $p$, is:

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

You can also use calculator functions such as `binompdf` and `binomcdf`.

### Mean and Standard Deviation of a Binomial Variable

If $X$ is binomial with $n$ trials and success probability $p$, then:

$$\mu_X = np$$

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

The mean is the expected number of successes in the [long run](/ap-stats/key-terms/long-run "fv-autolink"). The standard deviation tells you how much the count of successes typically varies from that [expected value](/ap-stats/key-terms/expected-value "fv-autolink").

## How to Use This on the AP Statistics Exam

### Problem Solving

1. Identify $n$, $p$, and what counts as a success.
2. Check BINS before using any formula.
3. Write the probability expression or [parameter](/ap-stats/key-terms/parameter "fv-autolink") formula clearly.
4. Give the answer in context.

### Common Trap

Watch the wording. "Exactly 3" means $P(X=3)$, "at most 3" means $P(X \le 3)$, and "at least 3" usually means using a [complement](/ap-stats/key-terms/complement "fv-autolink").

## Practice Example

Suppose 10 people each try a new snack and each person independently likes it with probability $0.5$. Let $X$ be the number who like the snack.

The probability that exactly 3 people like it is:

$$P(X=3)=\binom{10}{3}(0.5)^3(0.5)^7 \approx 0.117$$

The parameters are:

$$\mu = np = 10(0.5) = 5$$

$$\sigma = \sqrt{10(0.5)(0.5)} \approx 1.58$$

That means you expect about 5 of the 10 people to like the snack, and the count typically varies by about 1.58 people from that expected value.

## Common Misconceptions

- **Binomial counts successes, not trials.**
- **The formulas work before checking conditions.** They do not.
- **The standard deviation is $np(1-p)$.** That is the [variance](/ap-stats/key-terms/variance "fv-autolink"), not the standard deviation.
- **A success has to be something good.** It does not. "Success" is just the outcome you chose to count.

## Related AP Statistics Guides

- [Unit 2 Overview: Probability, Random Variables, and Probability Distributions](/ap-stats/unit-2/review/study-guide/JeaowG56kC80Eu94VWs7)
- [2.8 Introduction to Random Variables and Probability Distributions](/ap-stats/unit-2/intro-random-variables-probability-distributions/study-guide/B5MJ1YqQJ4D455wegCvz)
- [2.9 Parameters of Random Variables](/ap-stats/unit-2/mean-standard-deviation-random-variables/study-guide/g1wJm2o6V4wCoqbsC5Px)
- [2.11 The Normal Distribution](/ap-stats/unit-2/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66)

## Vocabulary

- **binomial distribution**: A probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success.
- **binomial probability function**: The formula P(X=x)=C(n,x)p^x(1-p)^(n-x) that calculates the probability of exactly x successes in n independent trials with probability of success p.
- **binomial random variable**: A random variable that counts the number of successes in a fixed number of repeated independent trials, where each trial has two possible outcomes.
- **independent trials**: Repeated experiments or observations where the outcome of one trial does not affect the outcome of any other trial.
- **number of failures**: The count of unfavorable outcomes in a sample, denoted as n(1-p̂), used to verify the normality condition.
- **number of successes**: The count of favorable outcomes in a sample, denoted as np̂, used to verify the normality condition.
- **probability distribution**: A function that describes the likelihood of all possible values of a random variable.
- **probability of success**: The constant probability p that an individual trial results in a success in a binomial experiment.
- **random number generator**: A tool or method used to randomly select items from a population for inclusion in a simple random sample.
- **simulation**: A method of modeling random events so that simulated outcomes closely match real-world outcomes, used to estimate probabilities.

## FAQs

### What is a binomial distribution in AP Stats?

A binomial distribution models the number of successes in a fixed number of independent trials when each trial has two outcomes and the probability of success stays constant.

### How do I know if a setting is binomial?

Check for a fixed number of trials, independent trials, two outcomes on each trial, and the same probability of success on every trial.

### What is the binomial formula for AP Statistics?

For exactly x successes in n independent trials with success probability p, use P(X=x)=C(n,x)p^x(1-p)^(n-x).

### When do I use binompdf vs binomcdf?

Use binompdf for exactly one value, such as P(X=3). Use binomcdf for at most a value, such as P(X≤3). For at least a value, use a complement when it is simpler.

### What does success mean in a binomial problem?

Success is just the outcome you choose to count. It does not have to be a good result; it could be a defective item, a correct answer, or a person with a certain trait.

### How should I set up a binomial FRQ answer?

Define X, identify n and p, verify the binomial conditions if needed, write the probability expression or calculator command, and interpret the final probability in context.

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