Introduction to the Binomial Distribution - AP Statistics
Card 1 of 30
Identify whether this is binomial: sample 5 cards without replacement, $X$ = number of aces.
Identify whether this is binomial: sample 5 cards without replacement, $X$ = number of aces.
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No; trials are not independent (without replacement). Removing cards changes probabilities between draws.
No; trials are not independent (without replacement). Removing cards changes probabilities between draws.
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Which option lists the four binomial conditions as $BINS$?
Which option lists the four binomial conditions as $BINS$?
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$BINS$: Binary outcomes, Independent trials, fixed Number of trials, Same $p$. Mnemonic for Binary, Independent, Number fixed, Same probability.
$BINS$: Binary outcomes, Independent trials, fixed Number of trials, Same $p$. Mnemonic for Binary, Independent, Number fixed, Same probability.
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What is the notation and meaning of $X \sim \text{Bin}(n,p)$?
What is the notation and meaning of $X \sim \text{Bin}(n,p)$?
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$X$ counts successes in $n$ trials with success probability $p$. Standard notation for binomial random variable.
$X$ counts successes in $n$ trials with success probability $p$. Standard notation for binomial random variable.
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State the probability formula for exactly $k$ successes: $P(X=k)$ for $X \sim \text{Bin}(n,p)$.
State the probability formula for exactly $k$ successes: $P(X=k)$ for $X \sim \text{Bin}(n,p)$.
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$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing $k$ spots with probability of exact outcome.
$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing $k$ spots with probability of exact outcome.
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What is the formula for the binomial coefficient $\binom{n}{k}$?
What is the formula for the binomial coefficient $\binom{n}{k}$?
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$\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts ways to choose $k$ items from $n$.
$\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts ways to choose $k$ items from $n$.
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What is $P(X=0)$ for $X \sim \text{Bin}(n,p)$?
What is $P(X=0)$ for $X \sim \text{Bin}(n,p)$?
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$(1-p)^n$. All trials fail when each has probability $(1-p)$.
$(1-p)^n$. All trials fail when each has probability $(1-p)$.
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Find $P(X\ge 2)$ for $X \sim \text{Bin}(6,0.1)$ using a complement.
Find $P(X\ge 2)$ for $X \sim \text{Bin}(6,0.1)$ using a complement.
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$1-[P(X=0)+P(X=1)]$. Use complement: $P(X≥2)=1-P(X≤1)$.
$1-[P(X=0)+P(X=1)]$. Use complement: $P(X≥2)=1-P(X≤1)$.
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Find $P(X=3)$ for $X \sim \text{Bin}(5,0.2)$.
Find $P(X=3)$ for $X \sim \text{Bin}(5,0.2)$.
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$\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.
$\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.
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What is the mean of $X \sim \text{Bin}(n,p)$?
What is the mean of $X \sim \text{Bin}(n,p)$?
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$\mu=np$. Expected successes equals trials times success probability.
$\mu=np$. Expected successes equals trials times success probability.
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Find $P(X\le 1)$ for $X \sim \text{Bin}(4,0.3)$ as a sum of exact probabilities.
Find $P(X\le 1)$ for $X \sim \text{Bin}(4,0.3)$ as a sum of exact probabilities.
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$\binom{4}{0}(0.3)^0(0.7)^4+\binom{4}{1}(0.3)^1(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.
$\binom{4}{0}(0.3)^0(0.7)^4+\binom{4}{1}(0.3)^1(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.
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Find $\mu$ and $\sigma$ for $X \sim \text{Bin}(20,0.4)$.
Find $\mu$ and $\sigma$ for $X \sim \text{Bin}(20,0.4)$.
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$\mu=8,\ \sigma=\sqrt{4.8}$. $\mu=20(0.4)=8$; $\sigma=\sqrt{20(0.4)(0.6)}=\sqrt{4.8}$.
$\mu=8,\ \sigma=\sqrt{4.8}$. $\mu=20(0.4)=8$; $\sigma=\sqrt{20(0.4)(0.6)}=\sqrt{4.8}$.
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What is the variance of $X \sim \text{Bin}(n,p)$?
What is the variance of $X \sim \text{Bin}(n,p)$?
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$\text{Var}(X)=np(1-p)$. Product of mean $np$ and failure probability $(1-p)$.
$\text{Var}(X)=np(1-p)$. Product of mean $np$ and failure probability $(1-p)$.
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Identify the distribution of the indicator variable $I$ for one trial with success probability $p$.
Identify the distribution of the indicator variable $I$ for one trial with success probability $p$.
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$I \sim \text{Bernoulli}(p)$. Single trial with two outcomes is Bernoulli.
$I \sim \text{Bernoulli}(p)$. Single trial with two outcomes is Bernoulli.
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Which option gives the range of possible values for $X \sim \text{Bin}(n,p)$?
Which option gives the range of possible values for $X \sim \text{Bin}(n,p)$?
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$X \in {0,1,2,\dots,n}$. Can have 0 to $n$ successes in $n$ trials.
$X \in {0,1,2,\dots,n}$. Can have 0 to $n$ successes in $n$ trials.
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Identify whether this is binomial: 10 coin flips, let $X$ be number of heads.
Identify whether this is binomial: 10 coin flips, let $X$ be number of heads.
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Yes; $X \sim \text{Bin}(10,0.5)$. Fixed trials, constant $p=0.5$, independent flips.
Yes; $X \sim \text{Bin}(10,0.5)$. Fixed trials, constant $p=0.5$, independent flips.
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Which option describes the key difference between geometric and binomial settings?
Which option describes the key difference between geometric and binomial settings?
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Binomial: fixed $n$; geometric: trials continue until first success. Binomial has predetermined trials; geometric stops at success.
Binomial: fixed $n$; geometric: trials continue until first success. Binomial has predetermined trials; geometric stops at success.
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Identify the missing binomial condition: outcomes are success or failure, $n$ fixed, $p$ constant, and ____.
Identify the missing binomial condition: outcomes are success or failure, $n$ fixed, $p$ constant, and ____.
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Trials are independent. Independence ensures constant $p$ across all trials.
Trials are independent. Independence ensures constant $p$ across all trials.
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What four conditions must hold for a setting to be binomial (BINS)?
What four conditions must hold for a setting to be binomial (BINS)?
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Binary outcomes, Independent trials, fixed Number $n$, Same $p$ each trial. These ensure each trial has two outcomes with constant probability.
Binary outcomes, Independent trials, fixed Number $n$, Same $p$ each trial. These ensure each trial has two outcomes with constant probability.
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Find $P(X\ge 1)$ for $X\sim\text{Bin}(6,p)$ in terms of $p$.
Find $P(X\ge 1)$ for $X\sim\text{Bin}(6,p)$ in terms of $p$.
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$1-(1-p)^6$. Use complement: $P(X\ge 1)=1-P(X=0)$.
$1-(1-p)^6$. Use complement: $P(X\ge 1)=1-P(X=0)$.
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Find $P(X\le 1)$ for $X\sim\text{Bin}(4,0.3)$ using binomial probabilities.
Find $P(X\le 1)$ for $X\sim\text{Bin}(4,0.3)$ using binomial probabilities.
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$\binom{4}{0}(0.7)^4+\binom{4}{1}(0.3)(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.
$\binom{4}{0}(0.7)^4+\binom{4}{1}(0.3)(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.
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Find $P(X=3)$ for $X\sim\text{Bin}(5,0.2)$.
Find $P(X=3)$ for $X\sim\text{Bin}(5,0.2)$.
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$\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.
$\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.
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Identify whether this is binomial: sample $20$ students without replacement from a class of $20$ and count left-handed.
Identify whether this is binomial: sample $20$ students without replacement from a class of $20$ and count left-handed.
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No; trials are not independent (without replacement from the entire class). Sampling entire population changes probabilities.
No; trials are not independent (without replacement from the entire class). Sampling entire population changes probabilities.
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Identify whether this is binomial: flip a fair coin $10$ times and count heads.
Identify whether this is binomial: flip a fair coin $10$ times and count heads.
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Yes; $X\sim\text{Bin}(10,0.5)$. Each flip is independent with $p=0.5$.
Yes; $X\sim\text{Bin}(10,0.5)$. Each flip is independent with $p=0.5$.
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What is the variance of $X\sim\text{Bin}(n,p)$?
What is the variance of $X\sim\text{Bin}(n,p)$?
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$\text{Var}(X)=np(1-p)$. Square of standard deviation; measures spread.
$\text{Var}(X)=np(1-p)$. Square of standard deviation; measures spread.
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What is the standard deviation of $X\sim\text{Bin}(n,p)$?
What is the standard deviation of $X\sim\text{Bin}(n,p)$?
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$\sigma=\sqrt{np(1-p)}$. Measures typical deviation from the mean.
$\sigma=\sqrt{np(1-p)}$. Measures typical deviation from the mean.
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What is the mean of a binomial random variable $X\sim\text{Bin}(n,p)$?
What is the mean of a binomial random variable $X\sim\text{Bin}(n,p)$?
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$\mu=np$. Expected number of successes in $n$ trials.
$\mu=np$. Expected number of successes in $n$ trials.
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State the formula for combinations $\binom{n}{k}$ using factorials.
State the formula for combinations $\binom{n}{k}$ using factorials.
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$\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts arrangements divided by duplicate orderings.
$\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts arrangements divided by duplicate orderings.
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What is the meaning of the notation $\binom{n}{k}$ in the binomial probability formula?
What is the meaning of the notation $\binom{n}{k}$ in the binomial probability formula?
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The number of ways to choose $k$ successes from $n$ trials. Also called "n choose k" or combinations.
The number of ways to choose $k$ successes from $n$ trials. Also called "n choose k" or combinations.
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State the probability formula for exactly $k$ successes when $X\sim\text{Bin}(n,p)$.
State the probability formula for exactly $k$ successes when $X\sim\text{Bin}(n,p)$.
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$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing positions with probability of outcomes.
$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing positions with probability of outcomes.
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What does the parameter $p$ represent in a binomial model $X\sim\text{Bin}(n,p)$?
What does the parameter $p$ represent in a binomial model $X\sim\text{Bin}(n,p)$?
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The probability of success on each trial. Remains constant across all trials in the experiment.
The probability of success on each trial. Remains constant across all trials in the experiment.
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