Introduction to the | AP Stats Unit 4 Study Guide

Quick Summary

This guide will equip you to master the binomial distribution, a fundamental tool for modeling discrete random variables. You will learn to identify the specific conditions that define a binomial setting, calculate the probability of a specific number of "successes" in a fixed number of trials, and determine the expected value (mean) and standard deviation for any binomial random variable.

Key Concepts

The binomial distribution is a specific type of discrete probability distribution. It's used when a random process has a fixed number of trials, and you're interested in counting the number of times a specific outcome, or "success," occurs.

The Four Conditions for a Binomial Setting (BINS)

For a random variable to be modeled by a binomial distribution, it must satisfy all four of the following conditions. Use the acronym BINS to remember them.

B - Binary: Each trial can be classified into one of two outcomes: success or failure.
- Example: A coin flip is either heads (success) or tails (failure). A free throw is either made (success) or missed (failure).
I - Independent: The outcome of one trial must not influence the outcome of any other trial.
- Example: Flipping a coin 10 times. The result of the first flip has no impact on the second.
- The 10% Condition: When sampling without replacement from a finite population, the trials are not technically independent. However, we can assume independence if the sample size, n, is no more than 10% of the population size, N (i.e., n \le 0.10N). This is a critical check!
N - Number of Trials: The number of trials, n, must be fixed in advance.
- Example: We will flip a coin exactly 20 times. We will survey exactly 100 students.
S - Same Probability: The probability of success, p, must be the same for each trial.
- Example: The probability of getting heads is 0.5 for every single coin flip.

The Binomial Random Variable and its Parameters

If a setting is binomial, we can define a binomial random variable, X, as the count of successes in n trials.

The two parameters that define a specific binomial distribution are:
- n: The fixed number of trials.
- p: The probability of success on any single trial.
We use the notation X ~ B(n, p) to say that "X is a binomial random variable with parameters n and p."

Calculating Binomial Probabilities

The probability of getting exactly k successes in n trials is given by the Binomial Probability Formula:

P(X = k) = (nCk) * p^k * (1-p)^(n-k)

Let's break this down:

k: The specific number of successes you are interested in.
(nCk): The binomial coefficient, read as "n choose k". It calculates the number of different ways you can arrange k successes among n trials.
- Formula: $n C k = n! / (k! * (n - k)!)$
- You will almost always use a calculator for this part.
p^k: The probability of getting k successes.
(1-p)^(n-k): The probability of getting (n-k) failures.

[Image: A histogram of a binomial distribution, B(20, 0.2). The x-axis is labeled "Number of Successes (k)" from 0 to 20. The y-axis is "Probability P(X=k)". The distribution is skewed to the right, with the highest bar at k=4.]

Mean and Standard Deviation of a Binomial Random Variable

For any binomial random variable X ~ B(n, p), you can calculate its center and spread using these simple formulas. You are expected to have these memorized.

Mean (Expected Value): μ_X = np
- This formula gives you the long-run average number of successes you would expect to see if you repeated the n trials many, many times.
Standard Deviation: σ_X = √[np(1-p)]
- This formula measures the typical distance of the outcomes (number of successes) from the mean. A larger standard deviation means more variability in the number of successes from one set of n trials to the next.

Key Vocabulary

Binomial Setting: A statistical experiment that satisfies the four BINS conditions (Binary, Independent, Number of trials fixed, Same probability of success).
Trial: A single repetition of a random process. For example, one coin flip, one free throw attempt, or one randomly selected person.
Success: The specific outcome of a trial that is being counted. Note that "success" doesn't have to be a positive outcome (e.g., "success" could be defined as a defective product).
Binomial Random Variable: A variable, X, that counts the number of successes in a binomial setting.
Binomial Coefficient (nCk): The number of ways to choose k successes from a set of n trials without regard to order.
Expected Value (Mean): The long-run average value of a random variable over many repetitions of the experiment. For a binomial variable, it's $n p$ .
Binomial Probability: The probability of getting exactly k successes in n trials in a binomial setting.

Calculator Tech (TI-84)

The TI-84 has powerful built-in functions for binomial calculations, found under the distribution menu: 2nd -> VARS [DISTR].

For "Exactly k" Successes: $bin o m p df ()$

Use this function to calculate P(X = k). The "pdf" stands for Probability Density Function.

Press 2nd -> VARS [DISTR].
Scroll down to A: binompdf(.
Enter the parameters:
- trials:n (the total number of trials)
- p:p (the probability of success)
- x value:k (the exact number of successes you want)
Select $P a s t e$ and press ENTER.

Example: To find the probability of getting exactly 7 heads in 10 coin flips: `binompdf(trials:10, p:0.5, x value:7) $returns $0.117$ .

For "At Most k" Successes: $bin o m c df ()$

Use this function to calculate P(X \le k). The "cdf" stands for Cumulative Distribution Function. It sums the probabilities from X=0 up to and including X=k.

Press 2nd -> VARS [DISTR].
Scroll down to B: binomcdf(.
Enter the parameters:
- trials:n
- p:p
- x value:k (the maximum number of successes)
Select $P a s t e$ and press ENTER.

Example: To find the probability of getting at most 7 heads in 10 coin flips, P(X \le 7): `binomcdf(trials:10, p:0.5, x value:7) $returns $0.945$ .

For "At Least k" Successes: $1 - bin o m c df ()$

The calculator cannot directly compute P(X \ge k). You must use the complement rule.

The complement of "at least k" successes is "at most k-1" successes.
Formula: P(X \ge k) = 1 - P(X \le k-1)

Example: To find the probability of getting at least 7 heads in 10 coin flips, P(X \ge 7):

This is equivalent to $1 - P (X \leq 6)$ .
Calculate 1 - binomcdf(trials:10, p:0.5, x value:6).
$1 - 0.828 = 0.172$ .

How to Show Work on the FRQ

To earn full credit for a binomial calculation on the AP exam, you must do more than just write down the calculator command. Follow this four-step process.

Template for Binomial Probability Calculations:

State the Distribution and Define the Variable:
- "Let X = the number of [describe successes] in a sample of [n trials]."
- "X follows a binomial distribution with parameters n = [value] and p = [value]."
Check Conditions (BINS):
- Binary: "Each trial is either a [success] or a [failure]."
- Independent: "The trials are independent because [state reason, e.g., 'the coin flips are independent' or 'the 10% condition is met since the sample size n=__ is less than 10% of the population of all __']."
- Number: "The number of trials is fixed at n = [value]."
- Same Probability: "The probability of success is constant at p = [value] for each trial."
Show the Calculation:
- Write the specific probability statement: e.g., $P (X = k)$ or $P (X \leq k)$ .
- Write the binomial formula with the correct numbers plugged in. This demonstrates your understanding.
- Example: $P (X = 7) = (10 C 7) * (0.5)^{7} * (0.5)^{3}$
State the Answer:
- Provide the final numerical answer, clearly labeled and in context.
- Example: "The probability of getting exactly 7 heads is approximately 0.117."

Template for Mean/Standard Deviation:

Identify Parameters: State the values of n and p.
State the Formula: Write the formula you are using (e.g., $μ_{X} = n p$ ).
Plug in Values and Calculate: Show the substitution (e.g., $μ_{X} = 10 * 0.5 = 5$ ).
Interpret in Context: Explain what the value means. "If we were to repeat this process of [flipping 10 coins] many times, the average number of [heads] we would expect is [5]."

Practice Problems

Problem 1:

A recent survey found that 22% of American adults have a tattoo. You randomly select 15 American adults. Let X be the number of adults in your sample who have a tattoo.

(a) Show that this is a binomial setting.

(b) What is the probability that exactly 4 of the selected adults have a tattoo?

Solution:

(a) Checking the BINS conditions:

Binary: Each adult either has a tattoo (success) or does not have a tattoo (failure).
Independent: The adults are randomly selected. Since the sample size (n=15) is far less than 10% of all American adults, we can assume the trials are independent.
Number: The number of trials is fixed at n = 15.
Same Probability: The probability that a randomly selected adult has a tattoo is constant at p = 0.22 for each person.

Since all four conditions are met, this is a binomial setting.

(b) Probability of exactly 4 successes:

State the Distribution and Define the Variable: Let X = the number of adults with a tattoo in a sample of 15. X follows a binomial distribution with n = 15 and p = 0.22.
Conditions: Checked in part (a).
Show the Calculation: We want to find P(X = 4).
$P (X = 4) = (15 C 4) * (0.22)^{4} * (1 - 0.22)^{1} 1$
State the Answer: Using a calculator, binompdf(trials:15, p:0.22, x value:4) = 0.2177. The probability that exactly 4 of the 15 selected adults have a tattoo is approximately 0.2177.

(c) Expected Value:

Identify Parameters: n = 15, p = 0.22.
State the Formula: $μ_{X} = n p$
Plug in Values and Calculate: $μ_{X} = 15 * 0.22 = 3.3$
Interpret in Context: If we were to repeatedly take random samples of 15 American adults, the long-run average number of adults with a tattoo per sample would be 3.3.

Problem 2:

A basketball player makes 80% of her free throws. Assume each free throw attempt is independent. In an upcoming game, she will attempt 12 free throws.

(a) What is the probability that she makes at most 9 of her free throws?

(b) What is the probability that she makes at least 10 of her free throws?

Solution:

Let X = the number of made free throws in 12 attempts. This is a binomial setting with n = 12 and p = 0.80.

(a) Probability of at most 9 free throws:

State the Goal: We need to find P(X \le 9).
Show the Calculation: This is a cumulative probability. We could write it as $P (X = 0) + P (X = 1) + ... + P (X = 9)$ . For our work, we can state the calculator command we will use.
State the Answer: Using a calculator, binomcdf(trials:12, p:0.80, x value:9) = 0.3907. The probability that she makes at most 9 of her 12 free throws is approximately 0.3907.

(b) Probability of at least 10 free throws:

State the Goal: We need to find P(X \ge 10).
Show the Calculation: We must use the complement rule.
$P (X \geq 10) = 1 - P (X \leq 9)$
State the Answer: Using the result from part (a) or recalculating:
1 - binomcdf(trials:12, p:0.80, x value:9) = 1 - 0.3907 = 0.6093.
The probability that she makes at least 10 of her 12 free throws is approximately 0.6093.

Common Mistakes to Avoid

Forgetting to Check BINS: Do not assume a setting is binomial. On an FRQ, you must explicitly check all four conditions to justify using binomial calculations. Pay special attention to the 10% condition for independence when sampling without replacement.
Confusing $bin o m p df$ and $bin o m c df$ : This is a critical error. Remember: pdf is for a single point (P(X=k)). cdf is for a cumulative range (P(X\lek)).
Incorrectly Calculating "At Least" Probabilities: The most common error with $bin o m c df$ is calculating P(X \ge k). Do not calculate $1 - bin o m c df (k)$ . The correct formula is $P (X \geq k) = 1 - P (X \leq k - 1)$ . Always subtract the probability of the values you want to exclude.
Misidentifying n and k: Be careful to distinguish between n (the total number of trials) and k (the specific number of successes you are interested in). Read the problem carefully.
Ignoring Context in Interpretations: When asked to interpret the mean or standard deviation, don't just state the number. Relate it back to the problem. For the mean, always use language like "the long-run average" or "we expect, on average..." to describe the value.

Introduction to the Binomial Distribution - AP Statistics Study Guide