AP Statistics Flashcards: Parameters For A Binomial Distribution

What this deck covers

This deck focuses on Parameters For A 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 a common criterion for using normal approximation in a binomial distribution?

Answer: $np \geq 10$ and $n(1-p) \geq 10$. Rule of thumb for normal approximation.

Flashcard 2: Identify the two parameters of a binomial distribution.

Answer: $n$ (number of trials) and $p$ (probability of success). These define the distribution completely.

Flashcard 3: What distribution is used when modeling the total number of successes in $n$ trials?

Answer: Binomial distribution. Models count of successes in fixed trials.

Flashcard 4: In a binomial distribution, what happens as $n$ increases with fixed $p$?

Answer: Distribution becomes more symmetric. Central Limit Theorem effect.

Flashcard 5: State the condition for a binomial setting regarding trials.

Answer: Trials must be independent. Outcome of one trial doesn't affect others.

Flashcard 6: What does the parameter $p$ represent in a binomial distribution?

Answer: The probability of success. Constant probability for each trial.

Flashcard 7: What is the probability of no successes in any binomial distribution?

Answer: $(1-p)^n$. All trials fail with probability $(1-p)$.

Flashcard 8: What is the range of possible values for $p$ in a binomial distribution?

Answer: $0 \leq p \leq 1$. Probability must be between 0 and 1 inclusive.

Flashcard 9: What is the range of possible values for $p$ in a binomial distribution?

Answer: $0 \leq p \leq 1$. Probability must be between 0 and 1 inclusive.

Flashcard 10: What does the parameter $n$ represent in a binomial distribution?

Answer: The number of trials. Fixed number of independent trials.

Flashcard 11: Choose the word that completes: A binomial distribution models the number of   in n trials.

Answer: successes. Counts favorable outcomes in fixed trials.

Flashcard 12: If $p=0.5$ and $n=8$, find the probability of exactly 4 successes.

Answer: $P(X=4) = 0.2734$. Using $\binom{8}{4}(0.5)^8 = 70 \times \frac{1}{256}$.

Flashcard 13: State the condition for a binomial setting regarding outcomes.

Answer: Each trial has two possible outcomes: success or failure. Binary outcomes define binomial setting.

Flashcard 14: If $n=5$ and $p=0.2$, what is the probability of exactly 1 success?

Answer: $P(X=1) = 0.4096$. Using $\binom{5}{1}(0.2)(0.8)^4 = 5 \times 0.2 \times 0.4096$.

Flashcard 15: Identify the distribution when $n=1$ in a binomial distribution.

Answer: Bernoulli distribution. Single trial binomial distribution.

Flashcard 16: Calculate $P(X=0)$ for $n=3$ and $p=0.7$ using the binomial probability formula.

Answer: $P(X=0) = 0.027$. Using $(1-0.7)^3 = (0.3)^3 = 0.027$.

Flashcard 17: What does the parameter $n$ represent in a binomial distribution?

Answer: The number of trials. Fixed number of independent trials.

Flashcard 18: What is the binomial coefficient in the binomial probability formula?

Answer: $\binom{n}{k}$. "Choose $k" from$n$ trials.

Flashcard 19: What is the binomial probability formula?

Answer: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combination term times probability terms.

Flashcard 20: What is the probability of no successes in any binomial distribution?

Answer: $(1-p)^n$. All trials fail with probability $(1-p)$.

Flashcard 21: Calculate the variance if $n=20$ and $p=0.3$ for a binomial distribution.

Answer: $Var(X) = 4.2$. Using $Var(X) = np(1-p) = 20 \times 0.3 \times 0.7 = 4.2$.

Flashcard 22: Identify the type of random variable a binomial distribution models.

Answer: Discrete random variable. Counts whole numbers of successes only.

Flashcard 23: State the condition for a binomial setting regarding trials.

Answer: Trials must be independent. Outcome of one trial doesn't affect others.

Flashcard 24: What is the shape of a binomial distribution when $p=0.5$?

Answer: Symmetrical. Equal probability of success and failure creates symmetry.

Flashcard 25: Find the probability of exactly 3 successes in 4 trials with $p=0.5$.

Answer: $P(X=3) = 0.25$. Using $\binom{4}{3}(0.5)^3(0.5)^1 = 4 \times 0.0625 = 0.25$.

Flashcard 26: In a binomial distribution, what happens as $n$ increases with fixed $p$?

Answer: Distribution becomes more symmetric. Central Limit Theorem effect.

Flashcard 27: What is the binomial coefficient in the binomial probability formula?

Answer: $\\binom{n}{k}$. "Choose $k$" from $n$ trials.

Flashcard 28: Find the probability of exactly 3 successes in 4 trials with $p=0.5$.

Answer: $P(X=3) = 0.25$. Using $\binom{4}{3}(0.5)^3(0.5)^1 = 4 \times 0.0625 = 0.25$.

Flashcard 29: What does the parameter $p$ represent in a binomial distribution?

Answer: The probability of success. Constant probability for each trial.

Flashcard 30: Determine the mean number of successes for $n=12$ and $p=0.25$.

Answer: $E(X) = 3$. Using $E(X) = np = 12 \times 0.25 = 3$.

Flashcard 31: Choose the word that completes: A binomial distribution models the number of   in n trials.

Answer: successes. Counts favorable outcomes in fixed trials.

Flashcard 32: State the condition for a binomial setting regarding probability.

Answer: Probability of success is constant for each trial. Same probability for every trial.

Flashcard 33: Identify the type of random variable a binomial distribution models.

Answer: Discrete random variable. Counts whole numbers of successes only.

Flashcard 34: What is the binomial probability formula?

Answer: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combination term times probability terms.

Flashcard 35: What is the shape of a binomial distribution when $p=0.5$?

Answer: Symmetrical. Equal probability of success and failure creates symmetry.

Flashcard 36: What happens to the distribution shape as $p$ approaches 0 or 1?

Answer: It becomes skewed. Extreme values create asymmetry.

Flashcard 37: What happens to the distribution shape as $p$ approaches 0 or 1?

Answer: It becomes skewed. Extreme values create asymmetry.

Flashcard 38: Calculate the standard deviation if $n=15$ and $p=0.6$.

Answer: $SD(X) = 1.89$. Using $SD(X) = \sqrt{15 \times 0.6 \times 0.4} = \sqrt{3.6} \approx 1.89$.

Flashcard 39: Calculate the standard deviation if $n=15$ and $p=0.6$.

Answer: $SD(X) = 1.89$. Using $SD(X) = \sqrt{15 \times 0.6 \times 0.4} = \sqrt{3.6} \approx 1.89$.

Flashcard 40: What is a common criterion for using normal approximation in a binomial distribution?

Answer: $np \geq 10$ and $n(1-p) \geq 10$. Rule of thumb for normal approximation.

Flashcard 41: Determine the mean number of successes for $n=12$ and $p=0.25$.

Answer: $E(X) = 3$. Using $E(X) = np = 12 \times 0.25 = 3$.

Flashcard 42: What is the formula for the expected value of a binomial distribution?

Answer: $E(X) = n \times p$. Mean of binomial distribution equals trials times probability.

Flashcard 43: Find the expected number of successes if $n=10$ and $p=0.5$.

Answer: $E(X) = 5$. Using $E(X) = np = 10 \times 0.5 = 5$.

Flashcard 44: What is the relationship between a binomial distribution and a normal distribution?

Answer: Normal approximation when $n$ is large and $p$ is not near 0 or 1. Normal approximation works for large $n$.

Flashcard 45: What is required for trials to be considered independent in a binomial setting?

Answer: Outcome of one trial does not affect another. No influence between separate trials.

Flashcard 46: What is the formula for the variance of a binomial distribution?

Answer: $Var(X) = n \times p \times (1-p)$. Variance formula includes the complement $(1-p)$ term.

Flashcard 47: What is the formula for the variance of a binomial distribution?

Answer: $Var(X) = n \times p \times (1-p)$. Variance formula includes the complement $(1-p)$ term.

Flashcard 48: If $n=5$ and $p=0.4$, calculate $P(X=2)$. Use binomial probability formula.

Answer: $P(X=2) = 0.3456$. Using $\binom{5}{2}(0.4)^2(0.6)^3 = 10 \times 0.16 \times 0.216$.

Flashcard 49: What is the formula for the standard deviation of a binomial distribution?

Answer: $SD(X) = \sqrt{n \times p \times (1-p)}$. Standard deviation is the square root of variance.

Flashcard 50: What is required for trials to be considered independent in a binomial setting?

Answer: Outcome of one trial does not affect another. No influence between separate trials.

Flashcard 51: Find the expected number of successes if $n=10$ and $p=0.5$.

Answer: $E(X) = 5$. Using $E(X) = np = 10 \times 0.5 = 5$.

Flashcard 52: State the condition for a binomial setting regarding outcomes.

Answer: Each trial has two possible outcomes: success or failure. Binary outcomes define binomial setting.

Flashcard 53: For $n=6$ and $p=0.3$, what is the probability of 0 successes?

Answer: $P(X=0) = 0.1176$. Using $(1-0.3)^6 = (0.7)^6 = 0.1176$.

Flashcard 54: What is the nature of individual events in a binomial distribution?

Answer: Mutually exclusive. Cannot occur simultaneously.

Flashcard 55: If $n=5$ and $p=0.2$, what is the probability of exactly 1 success?

Answer: $P(X=1) = 0.4096$. Using $\binom{5}{1}(0.2)(0.8)^4 = 5 \times 0.2 \times 0.4096$.

Flashcard 56: What distribution is used when modeling the total number of successes in $n$ trials?

Answer: Binomial distribution. Models count of successes in fixed trials.

Flashcard 57: What is the formula for the standard deviation of a binomial distribution?

Answer: $SD(X) = \sqrt{n \times p \times (1-p)}$. Standard deviation is the square root of variance.

Flashcard 58: How are binomial probabilities calculated?

Answer: Using the binomial probability formula. Uses combination and probability of success/failure.

Flashcard 59: State the condition for a binomial setting regarding probability.

Answer: Probability of success is constant for each trial. Same probability for every trial.

Flashcard 60: Calculate $P(X=0)$ for $n=3$ and $p=0.7$ using the binomial probability formula.

Answer: $P(X=0) = 0.027$. Using $(1-0.7)^3 = (0.3)^3 = 0.027$.

Flashcard 61: What is the formula for the expected value of a binomial distribution?

Answer: $E(X) = n \times p$. Mean of binomial distribution equals trials times probability.

Flashcard 62: Identify the two parameters of a binomial distribution.

Answer: $n$ (number of trials) and $p$ (probability of success). These define the distribution completely.

Flashcard 63: Calculate the variance if $n=20$ and $p=0.3$ for a binomial distribution.

Answer: $Var(X) = 4.2$. Using $Var(X) = np(1-p) = 20 \times 0.3 \times 0.7 = 4.2$.

Flashcard 64: How are binomial probabilities calculated?

Answer: Using the binomial probability formula. Uses combination and probability of success/failure.

Flashcard 65: Identify the distribution when $n=1$ in a binomial distribution.

Answer: Bernoulli distribution. Single trial binomial distribution.

Flashcard 66: If $p=0.5$ and $n=8$, find the probability of exactly 4 successes.

Answer: $P(X=4) = 0.2734$. Using $\binom{8}{4}(0.5)^8 = 70 \times \frac{1}{256}$.

Flashcard 67: If $n=5$ and $p=0.4$, calculate $P(X=2)$. Use binomial probability formula.

Answer: $P(X=2) = 0.3456$. Using $\binom{5}{2}(0.4)^2(0.6)^3 = 10 \times 0.16 \times 0.216$.

Flashcard 68: What is the nature of individual events in a binomial distribution?

Answer: Mutually exclusive. Cannot occur simultaneously.

Flashcard 69: What is the relationship between a binomial distribution and a normal distribution?

Answer: Normal approximation when $n$ is large and $p$ is not near 0 or 1. Normal approximation works for large $n$.