Sample Spaces and Events

Math for CS · Axiom Academy

Defining the universe of outcomes A sample space is the set of all possible outcomes of a random experiment. We denote it with the Greek letter Omega: Ω . Understanding sample spaces is crucial because probability theory is built upon them. Exhaustive: Every possible outcome must be included in Ω. Nothing can happen outside Ω. Mutually Exclusive: Outcomes in Ω are disjoint. If one outcome occurs, no other outcome can occur simultaneously. Finite or Infinite: A sample space can have finitely many outcomes (coin flip) or infinitely many (lifetime of a light bulb). Events: Subsets of the Sample Space An event is any subset of the sample space. It's a collection of outcomes we're interested in. Events are the "things we care about" in probability. Fair Die Roll: Ω = 1, 2, 3, 4, 5, 6 Two Coin Flips: Ω = HH, HT, TH, TT Empty Event (∅): The impossible event containing no outcomes. P(∅) = 0. Universal Event (Ω): The certain event containing all outcomes. P(Ω) = 1. Singleton Event: An event containing exactly one outcome, like 6 in a die roll. Just like sets, events can be combined using operations from set theory: Events A and B are disjoint (or mutually exclusive ) if they share no outcomes: A ∩ B = ∅. Complex sample spaces are often visualized using tree diagrams or tables: For experiments with multiple independent choices, use the multiplication principle: Finite vs. Infinite Sample Spaces

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