Basic Probability

GRE Math Subject · Axiom Academy

Sample spaces, events, conditional probability, independence, and Bayes' theorem A sample space Ω is the set of all possible outcomes of an experiment. Example: Rolling a die has sample space Ω = 1, 2, 3, 4, 5, 6 An event is a subset of the sample space. For rolling a die: Event A: "rolling an even number" = 2, 4, 6 Event B: "rolling greater than 3" = 4, 5, 6 Probability of an event (uniform sample space): For rolling an even: P(even) = 3/6 = 1/2 Conditional probability is the probability of event A given that event B has occurred. Example: In a deck of 52 cards, what's the probability of drawing an Ace given we drew a spade? Two events A and B are independent if the occurrence of one doesn't affect the probability of the other. Example: Flipping a coin twice First flip is heads: doesn't affect second flip Drawing without replacement: NOT independent (probabilities change) Drawing with replacement: Independent (probabilities stay the same) If events B₁, B₂, ..., Bₙ partition the sample space (mutually exclusive and exhaustive), then: Example: Testing for a disease P(disease) = 0.01, P(no disease) = 0.99 P(positive test | disease) = 0.95 P(positive test | no disease) = 0.05 P(positive) = 0.95 × 0.01 + 0.05 × 0.99 = 0.059 Bayes' Theorem allows us to update probabilities when we have new information. Example: Disease Testing (continued) If you test positive, what's the probability you actually have the disease?

This is the written version of the interactive lesson above. See the full GRE Math Subject course.