AP Statistics: Probability and Random Variables

This chapter establishes the mathematical framework for understanding chance and randomness. You will learn the fundamental rules of probability, starting with concepts like sample spaces, events, and complements. The curriculum then builds on this with rules for calculating the probability of combined events, including the General Addition Rule for 'or' scenarios and the Multiplication Rule for 'and' scenarios, which introduces the important idea of conditional probability and independence.

The second half of the chapter introduces random variables, which are variables whose numerical outcomes are determined by chance. You will learn to distinguish between discrete random variables (with countable outcomes) and continuous random variables (with outcomes in a range). For these variables, you will calculate and interpret their expected value (mean) and standard deviation. The chapter also covers how to determine the mean and variance when you combine or transform random variables.

Students are expected to master specific probability models, primarily the binomial and geometric distributions, and use their formulas to solve problems. The most challenging part is often correctly identifying which probability rule or distribution applies to a given scenario. A solid understanding of set notation and basic algebra from earlier studies is helpful, as the chapter's notation can be dense.

Formulas for AP Statistics: Probability and Random Variables

General Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ The probability that event A or event B (or both) will occur.
Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ The probability of event A occurring, given that event B has already occurred.
General Multiplication Rule: $P(A \cap B) = P(A) \cdot P(B|A)$ The probability that both event A and event B will occur.
Expected Value of a Discrete Random Variable: $E(X) = \mu_X = \sum x_i p_i$ A weighted average of the possible values of X, where p_i is the probability of value x_i.
Variance of a Discrete Random Variable: $\sigma_X^2 = \sum (x_i - \mu_X)^2 p_i$ The standard deviation is the square root of the variance, \sigma_X.
Mean of the Sum/Difference of Random Variables: $\mu_{X \pm Y} = \mu_X \pm \mu_Y$ The mean of a sum is the sum of the means; the mean of a difference is the difference of the means.
Variance of the Sum/Difference of Independent Random Variables: $\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2$ Applies only if X and Y are independent. Variances always add.
Binomial Probability Formula: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ Probability of k successes in n fixed, independent trials with success probability p.
Mean of a Binomial Distribution: $\mu_X = np$ For a binomial random variable X with parameters n and p.
Standard Deviation of a Binomial Distribution: $\sigma_X = \sqrt{np(1-p)}$ For a binomial random variable X with parameters n and p.
Geometric Probability Formula: $P(X=k) = (1-p)^{k-1} p$ Probability that the first success occurs on the k-th trial.
Mean of a Geometric Distribution: $\mu_X = \frac{1}{p}$ The expected number of trials required to get the first success.