AP Statistics Flashcards: Random Variables And Probability Distributions

What this deck covers

This deck focuses on Random Variables And Probability Distributions, 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: State the formula for the binomial probability.

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

Flashcard 2: Identify the type of random variable: number of heads in 10 coin flips.

Answer: Discrete random variable. Countable outcomes (0-10 heads).

Flashcard 3: Find the probability of getting at least one 6 in two dice rolls.

Answer: $\frac{11}{36}$. $1 - P(\text{no sixes}) = 1 - (\frac{5}{6})^2$

Flashcard 4: Determine the probability of drawing a red card from a standard deck.

Answer: $\frac{1}{2}$. 26 red cards in 52 total.

Flashcard 5: What is the probability of getting a sum of 7 when rolling two dice?

Answer: $\frac{1}{6}$. 6 ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

Flashcard 6: What is the probability of getting a sum of 7 when rolling two dice?

Answer: $\frac{1}{6}$. 6 ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

Flashcard 7: What is a random variable?

Answer: A variable whose values depend on outcomes of a random phenomenon. Its value is determined by chance events.

Flashcard 8: What is the mean of a probability distribution?

Answer: The expected value of the distribution. The central tendency of the distribution.

Flashcard 9: Which property must a PDF satisfy?

Answer: The total area under the curve is 1. Area represents total probability.

Flashcard 10: Calculate the expected value of a fair six-sided die roll.

Answer: $3.5$. $\frac{1+2+3+4+5+6}{6} = 3.5$

Flashcard 11: Identify the type of random variable: height of students in a class.

Answer: Continuous random variable. Can take any value in range.

Flashcard 12: What is the Poisson distribution?

Answer: A distribution representing the number of events in a fixed interval of time or space. Rate parameter $\lambda$ is key.

Flashcard 13: Which property must a PDF satisfy?

Answer: The total area under the curve is 1. Area represents total probability.

Flashcard 14: Determine the probability of drawing a red card from a standard deck.

Answer: $\frac{1}{2}$. 26 red cards in 52 total.

Flashcard 15: Find the variance of a fair six-sided die roll.

Answer: $\frac{35}{12}$. $E(X^2) - [E(X)]^2 = \frac{35}{12}$

Flashcard 16: What is the main characteristic of a geometric distribution?

Answer: Counts the number of trials until the first success. Memoryless property applies.

Flashcard 17: Find the probability of no heads in 3 coin flips.

Answer: $\frac{1}{8}$. $(\frac{1}{2})^3 = \frac{1}{8}$

Flashcard 18: Calculate the probability of drawing an ace from a standard deck of cards.

Answer: $\frac{1}{13}$. 4 aces in 52 cards.

Flashcard 19: Find the probability of rolling a 3 on a fair six-sided die.

Answer: $\frac{1}{6}$. Each outcome equally likely.

Flashcard 20: What is a probability distribution?

Answer: A function that gives the probabilities of different outcomes of a random variable. Maps outcomes to their probabilities.

Flashcard 21: What is a random variable?

Answer: A variable whose values depend on outcomes of a random phenomenon. Its value is determined by chance events.

Flashcard 22: What is the probability density function (PDF)?

Answer: A function that describes the likelihood of a continuous random variable to take on a value. For continuous variables; area gives probability.

Flashcard 23: What is a binomial distribution?

Answer: The distribution of the number of successes in a fixed number of independent Bernoulli trials. Fixed trials with constant success probability.

Flashcard 24: Define a continuous random variable.

Answer: A random variable with an infinite number of possible values in an interval. Can take any value within a range.

Flashcard 25: What does it mean if a random variable is normally distributed?

Answer: It follows a bell-shaped curve, symmetric about the mean. 68-95-99.7 rule applies.

Flashcard 26: Calculate the expected value of a fair six-sided die roll.

Answer: $3.5$. $\frac{1+2+3+4+5+6}{6} = 3.5$

Flashcard 27: What is the expected value of a uniform distribution over $[a, b]$?

Answer: The midpoint, $\frac{a+b}{2}$. Average of the endpoints.

Flashcard 28: Define a continuous random variable.

Answer: A random variable with an infinite number of possible values in an interval. Can take any value within a range.

Flashcard 29: What is a probability distribution?

Answer: A function that gives the probabilities of different outcomes of a random variable. Maps outcomes to their probabilities.

Flashcard 30: What is the standard deviation of a standard normal distribution?

Answer:

1. Standard normal is $N(0,1)$.

Flashcard 31: What is the mean of a standard normal distribution?

Answer:

1. Standard normal is $N(0,1)$.

Flashcard 32: Identify the property of a probability distribution function.

Answer: The sum of all probabilities is 1. All probabilities must add to 1.

Flashcard 33: What is the probability of success in a Bernoulli trial?

Answer: Denoted as $p$, where $0 \text{<} p \text{<} 1$. Complement is $(1-p)$.

Flashcard 34: What does it mean if a random variable is normally distributed?

Answer: It follows a bell-shaped curve, symmetric about the mean. 68-95-99.7 rule applies.

Flashcard 35: What is the cumulative distribution function (CDF)?

Answer: A function giving the probability that a random variable is less than or equal to a value. Cumulative probability up to a value.

Flashcard 36: What is the main characteristic of a geometric distribution?

Answer: Counts the number of trials until the first success. Memoryless property applies.

Flashcard 37: Determine the probability of drawing a heart from a standard deck of cards.

Answer: $\frac{1}{4}$. 13 hearts in 52 cards.

Flashcard 38: What is a binomial distribution?

Answer: The distribution of the number of successes in a fixed number of independent Bernoulli trials. Fixed trials with constant success probability.

Flashcard 39: Identify the type of random variable: number of heads in 10 coin flips.

Answer: Discrete random variable. Countable outcomes (0-10 heads).

Flashcard 40: Identify the key feature of a uniform distribution.

Answer: All outcomes are equally likely within the range. Constant probability density function.

Flashcard 41: Identify the main difference between binomial and hypergeometric distributions.

Answer: Binomial involves replacement; hypergeometric does not. Replacement changes the probability.

Flashcard 42: What is the cumulative distribution function (CDF)?

Answer: A function giving the probability that a random variable is less than or equal to a value. Cumulative probability up to a value.

Flashcard 43: What is the hypergeometric distribution?

Answer: Distribution of successes in draws without replacement from a finite population. Population size affects probability.

Flashcard 44: Find the probability of rolling a 3 on a fair six-sided die.

Answer: $\frac{1}{6}$. Each outcome equally likely.

Flashcard 45: Find the variance of a fair six-sided die roll.

Answer: $\frac{35}{12}$. $E(X^2) - [E(X)]^2 = \frac{35}{12}$

Flashcard 46: Calculate the probability of drawing an ace from a standard deck of cards.

Answer: $\frac{1}{13}$. 4 aces in 52 cards.

Flashcard 47: Define a discrete random variable.

Answer: A random variable with a countable number of possible values. Can take values like 1, 2, 3, etc.

Flashcard 48: Find the probability of getting at least one 6 in two dice rolls.

Answer: $\frac{11}{36}$. $1 - P(\text{no sixes}) = 1 - (\frac{5}{6})^2$

Flashcard 49: Define the probability mass function (PMF).

Answer: A function giving the probability that a discrete random variable is exactly equal to some value. For discrete variables only.

Flashcard 50: Identify the main difference between binomial and hypergeometric distributions.

Answer: Binomial involves replacement; hypergeometric does not. Replacement changes the probability.

Flashcard 51: Find the probability of no heads in 3 coin flips.

Answer: $\frac{1}{8}$. $(\frac{1}{2})^3 = \frac{1}{8}$

Flashcard 52: Identify the type of random variable: height of students in a class.

Answer: Continuous random variable. Can take any value in range.

Flashcard 53: Determine the probability of drawing a heart from a standard deck of cards.

Answer: $\frac{1}{4}$. 13 hearts in 52 cards.

Flashcard 54: Define a Bernoulli random variable.

Answer: A random variable with two possible outcomes: success or failure. Simplest discrete distribution.

Flashcard 55: Define a discrete random variable.

Answer: A random variable with a countable number of possible values. Can take values like 1, 2, 3, etc.

Flashcard 56: What is the Poisson distribution?

Answer: A distribution representing the number of events in a fixed interval of time or space. Rate parameter $\lambda$ is key.

Flashcard 57: State the formula for the binomial probability.

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

Flashcard 58: What is the probability density function (PDF)?

Answer: A function that describes the likelihood of a continuous random variable to take on a value. For continuous variables; area gives probability.

Flashcard 59: What is the probability of success in a Bernoulli trial?

Answer: Denoted as $p$, where $0 \text{<} p \text{<} 1$. Complement is $(1-p)$.

Flashcard 60: Define a Bernoulli random variable.

Answer: A random variable with two possible outcomes: success or failure. Simplest discrete distribution.

Flashcard 61: What is the mean of a probability distribution?

Answer: The expected value of the distribution. The central tendency of the distribution.

Flashcard 62: Identify the property of a probability distribution function.

Answer: The sum of all probabilities is 1. All probabilities must add to 1.

Flashcard 63: What is the hypergeometric distribution?

Answer: Distribution of successes in draws without replacement from a finite population. Population size affects probability.

Flashcard 64: Identify the key feature of a uniform distribution.

Answer: All outcomes are equally likely within the range. Constant probability density function.

Flashcard 65: What is the expected value of a uniform distribution over $[a, b]$?

Answer: The midpoint, $\frac{a+b}{2}$. Average of the endpoints.

Flashcard 66: Define the probability mass function (PMF).

Answer: A function giving the probability that a discrete random variable is exactly equal to some value. For discrete variables only.