Parameters for a Binomial Distribution - AP Statistics
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What is the nature of individual events in a binomial distribution?
What is the nature of individual events in a binomial distribution?
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Mutually exclusive. Cannot occur simultaneously.
Mutually exclusive. Cannot occur simultaneously.
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If $n=5$ and $p=0.4$, calculate $P(X=2)$. Use binomial probability formula.
If $n=5$ and $p=0.4$, calculate $P(X=2)$. Use binomial probability formula.
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$P(X=2) = 0.3456$. Using $\binom{5}{2}(0.4)^2(0.6)^3 = 10 \times 0.16 \times 0.216$.
$P(X=2) = 0.3456$. Using $\binom{5}{2}(0.4)^2(0.6)^3 = 10 \times 0.16 \times 0.216$.
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What is the probability of no successes in any binomial distribution?
What is the probability of no successes in any binomial distribution?
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$(1-p)^n$. All trials fail with probability $(1-p)$.
$(1-p)^n$. All trials fail with probability $(1-p)$.
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What is the shape of a binomial distribution when $p=0.5$?
What is the shape of a binomial distribution when $p=0.5$?
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Symmetrical. Equal probability of success and failure creates symmetry.
Symmetrical. Equal probability of success and failure creates symmetry.
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Identify the distribution when $n=1$ in a binomial distribution.
Identify the distribution when $n=1$ in a binomial distribution.
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Bernoulli distribution. Single trial binomial distribution.
Bernoulli distribution. Single trial binomial distribution.
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Calculate the standard deviation if $n=15$ and $p=0.6$.
Calculate the standard deviation if $n=15$ and $p=0.6$.
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$SD(X) = 1.89$. Using $SD(X) = \sqrt{15 \times 0.6 \times 0.4} = \sqrt{3.6} \approx 1.89$.
$SD(X) = 1.89$. Using $SD(X) = \sqrt{15 \times 0.6 \times 0.4} = \sqrt{3.6} \approx 1.89$.
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Calculate the variance if $n=20$ and $p=0.3$ for a binomial distribution.
Calculate the variance if $n=20$ and $p=0.3$ for a binomial distribution.
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$Var(X) = 4.2$. Using $Var(X) = np(1-p) = 20 \times 0.3 \times 0.7 = 4.2$.
$Var(X) = 4.2$. Using $Var(X) = np(1-p) = 20 \times 0.3 \times 0.7 = 4.2$.
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Find the expected number of successes if $n=10$ and $p=0.5$.
Find the expected number of successes if $n=10$ and $p=0.5$.
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$E(X) = 5$. Using $E(X) = np = 10 \times 0.5 = 5$.
$E(X) = 5$. Using $E(X) = np = 10 \times 0.5 = 5$.
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Choose the word that completes: A binomial distribution models the number of _____ in n trials.
Choose the word that completes: A binomial distribution models the number of _____ in n trials.
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successes. Counts favorable outcomes in fixed trials.
successes. Counts favorable outcomes in fixed trials.
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What happens to the distribution shape as $p$ approaches 0 or 1?
What happens to the distribution shape as $p$ approaches 0 or 1?
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It becomes skewed. Extreme values create asymmetry.
It becomes skewed. Extreme values create asymmetry.
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Determine the mean number of successes for $n=12$ and $p=0.25$.
Determine the mean number of successes for $n=12$ and $p=0.25$.
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$E(X) = 3$. Using $E(X) = np = 12 \times 0.25 = 3$.
$E(X) = 3$. Using $E(X) = np = 12 \times 0.25 = 3$.
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If $p=0.5$ and $n=8$, find the probability of exactly 4 successes.
If $p=0.5$ and $n=8$, find the probability of exactly 4 successes.
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$P(X=4) = 0.2734$. Using $\binom{8}{4}(0.5)^8 = 70 \times \frac{1}{256}$.
$P(X=4) = 0.2734$. Using $\binom{8}{4}(0.5)^8 = 70 \times \frac{1}{256}$.
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What distribution is used when modeling the total number of successes in $n$ trials?
What distribution is used when modeling the total number of successes in $n$ trials?
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Binomial distribution. Models count of successes in fixed trials.
Binomial distribution. Models count of successes in fixed trials.
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Find the probability of exactly 3 successes in 4 trials with $p=0.5$.
Find the probability of exactly 3 successes in 4 trials with $p=0.5$.
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$P(X=3) = 0.25$. Using $\binom{4}{3}(0.5)^3(0.5)^1 = 4 \times 0.0625 = 0.25$.
$P(X=3) = 0.25$. Using $\binom{4}{3}(0.5)^3(0.5)^1 = 4 \times 0.0625 = 0.25$.
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Calculate $P(X=0)$ for $n=3$ and $p=0.7$ using the binomial probability formula.
Calculate $P(X=0)$ for $n=3$ and $p=0.7$ using the binomial probability formula.
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$P(X=0) = 0.027$. Using $(1-0.7)^3 = (0.3)^3 = 0.027$.
$P(X=0) = 0.027$. Using $(1-0.7)^3 = (0.3)^3 = 0.027$.
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What is a common criterion for using normal approximation in a binomial distribution?
What is a common criterion for using normal approximation in a binomial distribution?
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$np \geq 10$ and $n(1-p) \geq 10$. Rule of thumb for normal approximation.
$np \geq 10$ and $n(1-p) \geq 10$. Rule of thumb for normal approximation.
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What is the relationship between a binomial distribution and a normal distribution?
What is the relationship between a binomial distribution and a normal distribution?
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Normal approximation when $n$ is large and $p$ is not near 0 or 1. Normal approximation works for large $n$.
Normal approximation when $n$ is large and $p$ is not near 0 or 1. Normal approximation works for large $n$.
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Identify the type of random variable a binomial distribution models.
Identify the type of random variable a binomial distribution models.
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Discrete random variable. Counts whole numbers of successes only.
Discrete random variable. Counts whole numbers of successes only.
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In a binomial distribution, what happens as $n$ increases with fixed $p$?
In a binomial distribution, what happens as $n$ increases with fixed $p$?
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Distribution becomes more symmetric. Central Limit Theorem effect.
Distribution becomes more symmetric. Central Limit Theorem effect.
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What is required for trials to be considered independent in a binomial setting?
What is required for trials to be considered independent in a binomial setting?
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Outcome of one trial does not affect another. No influence between separate trials.
Outcome of one trial does not affect another. No influence between separate trials.
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State the condition for a binomial setting regarding probability.
State the condition for a binomial setting regarding probability.
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Probability of success is constant for each trial. Same probability for every trial.
Probability of success is constant for each trial. Same probability for every trial.
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What is the binomial probability formula?
What is the binomial probability formula?
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$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combination term times probability terms.
$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combination term times probability terms.
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How are binomial probabilities calculated?
How are binomial probabilities calculated?
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Using the binomial probability formula. Uses combination and probability of success/failure.
Using the binomial probability formula. Uses combination and probability of success/failure.
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State the condition for a binomial setting regarding outcomes.
State the condition for a binomial setting regarding outcomes.
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Each trial has two possible outcomes: success or failure. Binary outcomes define binomial setting.
Each trial has two possible outcomes: success or failure. Binary outcomes define binomial setting.
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State the condition for a binomial setting regarding trials.
State the condition for a binomial setting regarding trials.
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Trials must be independent. Outcome of one trial doesn't affect others.
Trials must be independent. Outcome of one trial doesn't affect others.
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What does the parameter $p$ represent in a binomial distribution?
What does the parameter $p$ represent in a binomial distribution?
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The probability of success. Constant probability for each trial.
The probability of success. Constant probability for each trial.
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What does the parameter $n$ represent in a binomial distribution?
What does the parameter $n$ represent in a binomial distribution?
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The number of trials. Fixed number of independent trials.
The number of trials. Fixed number of independent trials.
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Identify the two parameters of a binomial distribution.
Identify the two parameters of a binomial distribution.
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$n$ (number of trials) and $p$ (probability of success). These define the distribution completely.
$n$ (number of trials) and $p$ (probability of success). These define the distribution completely.
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What is the formula for the variance of a binomial distribution?
What is the formula for the variance of a binomial distribution?
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$Var(X) = n \times p \times (1-p)$. Variance formula includes the complement $(1-p)$ term.
$Var(X) = n \times p \times (1-p)$. Variance formula includes the complement $(1-p)$ term.
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If $n=5$ and $p=0.2$, what is the probability of exactly 1 success?
If $n=5$ and $p=0.2$, what is the probability of exactly 1 success?
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$P(X=1) = 0.4096$. Using $\binom{5}{1}(0.2)(0.8)^4 = 5 \times 0.2 \times 0.4096$.
$P(X=1) = 0.4096$. Using $\binom{5}{1}(0.2)(0.8)^4 = 5 \times 0.2 \times 0.4096$.
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