UnofficialAP StatisticsUnit 4 of 9

AP Statistics Unit 4: Probability, Random Variables and Probability Distributions

Probability rules, random variables, binomial and geometric distributions.

Unit overview

Probability describes long-run relative frequency. The rules you need are the complement rule, the addition rule $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and the multiplication rule $P(A \cap B) = P(A) \cdot P(B \mid A)$ . Events are independent when $P(B \mid A) = P(B)$ .

A random variable assigns a number to each outcome. Its mean $\mu_X = \sum x \, p(x)$ is the long-run average and its standard deviation measures typical distance from that mean. Binomial settings have a fixed number of independent trials with the same success probability; geometric settings count trials until the first success.

Key topics

Probability rules: Complement, addition and multiplication rules, two-way tables and tree diagrams.
Conditional probability: $P(A \mid B) = P(A \cap B)/P(B)$ and checking independence.
Discrete random variables: Mean, variance and standard deviation; linear transformations and sums of independent variables.
Binomial and geometric distributions: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ and $P(X = k) = (1-p)^{k-1} p$ .

Worked example

A free-throw shooter makes 70% of shots. Assuming independence, what is the probability she makes exactly 3 of 4 shots?

This is binomial with $n = 4$ , $p = 0.7$ , $k = 3$ .
$P(X = 3) = \binom{4}{3}(0.7)^3(0.3)^1 = 4 \cdot 0.343 \cdot 0.3$ .
$P(X = 3) = 0.4116$ .

Answer: $P(X = 3) = 0.4116$

Common mistakes

Adding probabilities of events that are not mutually exclusive without subtracting the overlap.
Assuming independence just because events seem unrelated; check with the definition or justify it from the design.
Adding standard deviations of independent variables instead of adding variances.

Exam tips

Define the random variable and state the distribution and its parameters before calculating; this earns communication credit.
For at least one questions, use the complement: $P(\text{at least one}) = 1 - P(\text{none})$ .

Descriptive Statistics Study Guide (study guide)
Statistics Formulas cheat sheet (one-page formulas)
Descriptive Statistics Quiz (quick quiz)
Statistics practice problems (with worked solutions)

See the AP Statistics exam format and every unit on the course page.