Probability: every key term you need (+ practice quiz)

More AP Statistics guides

• Designing Studies
• Exploring Bivariate Data
• Random Variables & Distributions
• Sampling Distributions
• Inference for Means

Probability Basics

Probability = likelihood of event occurring; 0 ≤ P(A) ≤ 1. P(A)=0 impossible, P(A)=1 certain. Frequency approach: P(A) ≈ n(A)/n as n→∞.

Sample Space

Set of all possible outcomes. For coin flip: {H,T}. For die: {1,2,3,4,5,6}. Exhaustive and mutually exclusive.

Complement Rule

P(Ac) = 1 - P(A). Probability of NOT A equals one minus probability of A. Useful for 'at least one' problems.

Addition Rule

P(A∪B) = P(A) + P(B) - P(A∩B). Subtract intersection to avoid double-counting.

Mutually Exclusive Events

A∩B = ∅, so P(A∩B)=0. Then P(A∪B) = P(A) + P(B). Events can't happen simultaneously.

Conditional Probability

P(A|B) = P(A∩B)/P(B); probability of A given B occurred. Requires P(B)≠0.

Independence

Events A and B independent if P(A|B) = P(A); knowledge of B doesn't affect probability of A.

Multiplication Rule

P(A∩B) = P(A)·P(B|A). For independent events: P(A∩B) = P(A)·P(B).

Bayes' Theorem

P(A|B) = P(B|A)·P(A) / P(B). Reverses conditional probability; useful for diagnostic testing.

Counting: Permutations

nPr = n!/(n-r)!; order matters. Arranging r objects from n; same objects in different order = different permutation.

Counting: Combinations

nCr = n!/(r!(n-r)!); order doesn't matter. Choosing r objects from n; same objects = same combination.

Binomial Coefficients

C(n,k) = (n choose k); appears in binomial expansion and probability calculations.

Tree Diagrams

Visual representation of sequential events; branches show outcomes at each stage. Useful for conditional probability.

Two-Way Tables for Probability

Rows and columns represent categorical variables. Use to calculate marginal and conditional probabilities.

Law of Total Probability

P(A) = Σ P(A|Bi)·P(Bi) over all partitions Bi. Condition on all possibilities.

Probability Distributions

List of all possible outcomes and their probabilities; must sum to 1. Can be discrete or continuous.

Unit 3 Summary

Probability rules enable calculating complex probabilities. Conditional probability and independence are key. Counting principles solve combinatorial problems.

Law of Large Numbers

As the number of trials increases, the observed proportion of an outcome converges to its true probability. It does NOT say short runs balance out.

Simulation Steps

State the question, describe how to model one repetition with a chance device (digits, coin, RNG), perform many repetitions, use the results to answer.

General Addition Rule

P(A or B) = P(A) + P(B) - P(A and B). Subtracting the overlap avoids double counting; for mutually exclusive events the overlap is 0.

General Multiplication Rule

P(A and B) = P(A) · P(B | A). Reduces to P(A)·P(B) only when A and B are independent.

Conditional Probability

P(B | A) = P(A and B) / P(A). In a two-way table: restrict to the row/column of the given condition, then take the fraction.

Independence Test

A and B are independent if P(B | A) = P(B), equivalently P(A and B) = P(A)·P(B). Check with numbers, not intuition.

Mutually Exclusive vs Independent

Disjoint events with nonzero probabilities can NEVER be independent: if A happens, B cannot, so P(B | A) = 0 ≠ P(B).

Complement Shortcut

P(at least one) = 1 - P(none). For independent trials with success probability p, P(at least one in n) = 1 - (1-p)^n.

Tree Diagram

Branches show sequential probabilities; multiply along a path for joint probability, add paths for a total. Ideal for conditional problems and Bayes-type reversals.

Bayes-Style Reversal

P(A | B) = P(A and B)/P(B), where P(B) sums all paths ending in B. Used for 'given a positive test, probability of disease' problems.

Sensitivity and False Positive

Sensitivity = P(positive | disease). False positive rate = P(positive | no disease). Rare diseases give low P(disease | positive) even with good tests.

Venn Diagram Regions

Fill the intersection first, then the 'only' regions by subtraction, then the outside region as 1 minus everything.

Sample Space

The set of all possible outcomes. Probabilities of all outcomes in the sample space sum to 1; each is between 0 and 1.

Probability Model

A sample space plus a probability assignment. Valid only if every probability is in [0,1] and they sum to 1.

Two-Way Table Probability

Joint = cell / grand total; marginal = row or column total / grand total; conditional = cell / row (or column) total.

Without Replacement Dependence

Draws without replacement are dependent; the second probability depends on the first result. P(both red) = (r/N)·((r-1)/(N-1)).

Gambler's Fallacy

Believing a 'due' outcome is more likely after a streak. Independent trials have no memory; probabilities do not change.

Random Digit Simulation

Assign digits to outcomes proportional to probability (e.g., 0-2 = success for p = 0.3, 3-9 = failure) and read digits one repetition at a time.

Long-Run Relative Frequency

Probability is the proportion of times an outcome would occur in a very long series of repetitions, not what must happen in a few trials.

Union vs Intersection Language

'Or' means union (A ∪ B, either or both). 'And' means intersection (A ∩ B, both). 'Given' means condition (|).