AP Statistics Flashcards: Introduction To The Binomial Distribution

What this deck covers

This deck focuses on Introduction To The Binomial Distribution, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $P(X=n)$ for $X \sim \text{Bin}(n,p)$?

Answer: $p^n$. All trials succeed when each has probability $p$.

Flashcard 2: Which option correctly interprets $P(X=0)$ for $X\sim\text{Bin}(n,p)$?

Answer: No successes in $n$ trials: $P(X=0)=(1-p)^n$. All failures occur with probability $(1-p)^n$.

Flashcard 3: What is the formula for the binomial coefficient $\binom{n}{k}$?

Answer: $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts ways to choose $k$ items from $n$.

Flashcard 4: Find $\mu$ and $\sigma$ for $X\sim\text{Bin}(25,0.4)$.

Answer: $\mu=10,\ \sigma=\sqrt{6}$. Use $\mu=np=25(0.4)$ and $\sigma=\sqrt{25(0.4)(0.6)}$.

Flashcard 5: Find $P(X\ge 1)$ for $X\sim\text{Bin}(6,p)$ in terms of $p$.

Answer: $1-(1-p)^6$. Use complement: $P(X\ge 1)=1-P(X=0)$.

Flashcard 6: What is the mean of a binomial random variable $X\sim\text{Bin}(n,p)$?

Answer: $\mu=np$. Expected number of successes in $n$ trials.

Flashcard 7: Identify whether this is binomial: sample 5 cards without replacement, $X$ = number of aces.

Answer: No; trials are not independent (without replacement). Removing cards changes probabilities between draws.

Flashcard 8: Which option lists the four binomial conditions as $BINS$?

Answer: $BINS$: Binary outcomes, Independent trials, fixed Number of trials, Same $p$. Mnemonic for Binary, Independent, Number fixed, Same probability.

Flashcard 9: What four conditions must hold for a setting to be binomial (BINS)?

Answer: Binary outcomes, Independent trials, fixed Number $n$, Same $p$ each trial. These ensure each trial has two outcomes with constant probability.

Flashcard 10: Find $P(X\le 1)$ for $X \sim \text{Bin}(4,0.3)$ as a sum of exact probabilities.

Answer: $\binom{4}{0}(0.3)^0(0.7)^4+\binom{4}{1}(0.3)^1(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.

Flashcard 11: Identify whether this is binomial: 10 coin flips, let $X$ be number of heads.

Answer: Yes; $X \sim \text{Bin}(10,0.5)$. Fixed trials, constant $p=0.5$, independent flips.

Flashcard 12: Identify whether this is binomial: sample $20$ students without replacement from a class of $20$ and count left-handed.

Answer: No; trials are not independent (without replacement from the entire class). Sampling entire population changes probabilities.

Flashcard 13: Identify the missing binomial condition: outcomes are success or failure, $n$ fixed, $p$ constant, and  .

Answer: Trials are independent. Independence ensures constant $p$ across all trials.

Flashcard 14: State the probability formula for exactly $k$ successes when $X\sim\text{Bin}(n,p)$.

Answer: $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing positions with probability of outcomes.

Flashcard 15: What is the standard deviation of $X\sim\text{Bin}(n,p)$?

Answer: $\sigma=\sqrt{np(1-p)}$. Measures typical deviation from the mean.

Flashcard 16: Which option describes the key difference between geometric and binomial settings?

Answer: Binomial: fixed $n$; geometric: trials continue until first success. Binomial has predetermined trials; geometric stops at success.

Flashcard 17: Find $P(X\ge 2)$ for $X \sim \text{Bin}(6,0.1)$ using a complement.

Answer: $1-[P(X=0)+P(X=1)]$. Use complement: $P(X≥2)=1-P(X≤1)$.

Flashcard 18: What does the parameter $p$ represent in a binomial model $X\sim\text{Bin}(n,p)$?

Answer: The probability of success on each trial. Remains constant across all trials in the experiment.

Flashcard 19: Which option gives the range of possible values for $X \sim \text{Bin}(n,p)$?

Answer: $X \in \{0,1,2,\dots,n\}$. Can have 0 to $n$ successes in $n$ trials.

Flashcard 20: What is $P(X=0)$ for $X \sim \text{Bin}(n,p)$?

Answer: $(1-p)^n$. All trials fail when each has probability $(1-p)$.

Flashcard 21: What is the support (possible values) of $X$ when $X\sim\text{Bin}(n,p)$?

Answer: Integers $0,1,2,\dots,n$. Can have anywhere from zero to all successes.

Flashcard 22: Identify the distribution of the number of failures $Y=n-X$ when $X\sim\text{Bin}(n,p)$.

Answer: $Y\sim\text{Bin}(n,1-p)$. Failures have probability $1-p$ each.

Flashcard 23: Find $P(X=3)$ for $X\sim\text{Bin}(5,0.2)$.

Answer: $\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.

Flashcard 24: What is the mean of $X \sim \text{Bin}(n,p)$?

Answer: $\mu=np$. Expected successes equals trials times success probability.

Flashcard 25: What is the variance of $X \sim \text{Bin}(n,p)$?

Answer: $\text{Var}(X)=np(1-p)$. Product of mean $np$ and failure probability $(1-p)$.

Flashcard 26: What does the parameter $n$ represent in a binomial model $X\sim\text{Bin}(n,p)$?

Answer: The fixed number of trials. How many times the experiment is repeated.

Flashcard 27: What is the variance of $X\sim\text{Bin}(n,p)$?

Answer: $\text{Var}(X)=np(1-p)$. Square of standard deviation; measures spread.

Flashcard 28: Find $\mu$ and $\sigma$ for $X \sim \text{Bin}(20,0.4)$.

Answer: $\mu=8,\ \sigma=\sqrt{4.8}$. $\mu=20(0.4)=8$; $\sigma=\sqrt{20(0.4)(0.6)}=\sqrt{4.8}$.

Flashcard 29: State the probability formula for exactly $k$ successes: $P(X=k)$ for $X \sim \text{Bin}(n,p)$.

Answer: $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing $k$ spots with probability of exact outcome.

Flashcard 30: State the formula for combinations $\binom{n}{k}$ using factorials.

Answer: $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts arrangements divided by duplicate orderings.

Flashcard 31: Find $P(X\le 1)$ for $X\sim\text{Bin}(4,0.3)$ using binomial probabilities.

Answer: $\binom{4}{0}(0.7)^4+\binom{4}{1}(0.3)(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.

Flashcard 32: What is the binomial distribution used to model in AP Statistics?

Answer: The number of successes $X$ in $n$ independent trials with constant $p$. Counts how many times an event occurs in repeated independent trials.

Flashcard 33: Identify whether this is binomial: flip a fair coin $10$ times and count heads.

Answer: Yes; $X\sim\text{Bin}(10,0.5)$. Each flip is independent with $p=0.5$.

Flashcard 34: What is the standard deviation of $X \sim \text{Bin}(n,p)$?

Answer: $\sigma=\sqrt{np(1-p)}$. Square root of variance $np(1-p)$.

Flashcard 35: Identify the complement event to "at least $k$ successes" for a binomial $X$.

Answer: "At most $k-1$ successes," i.e., $X\le k-1$. "At least $k$" means $X\ge k$; complement is $X<k$.

Flashcard 36: Identify the distribution of the indicator variable $I$ for one trial with success probability $p$.

Answer: $I \sim \text{Bernoulli}(p)$. Single trial with two outcomes is Bernoulli.

Flashcard 37: What is the meaning of the notation $\binom{n}{k}$ in the binomial probability formula?

Answer: The number of ways to choose $k$ successes from $n$ trials. Also called "n choose k" or combinations.

Flashcard 38: What is the notation and meaning of $X \sim \text{Bin}(n,p)$?

Answer: $X$ counts successes in $n$ trials with success probability $p$. Standard notation for binomial random variable.

Flashcard 39: Find $P(X=3)$ for $X \sim \text{Bin}(5,0.2)$.

Answer: $\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.