Learn/Combinatorics/Sample Spaces
Combinatorics • Topic 18

Sample Spaces

Probability begins with identifying all possible outcomes of an experiment.

Definitions

Definition (Sample Space)
The sample space ( or ) is the set of all possible outcomes of a random experiment.
Definition (Event)
An event () is a subset of the sample space ().

Constructing Sample Spaces

Identifying correctly is critical. We often use:
  1. Lists: For small spaces (e.g., coin flips ).
  2. Tree Diagrams: For multi-stage experiments.
  3. Grid/Coordinate Systems: For two independent variables (e.g., two dice).

Discrete vs Continuous

  • Discrete: Countable outcomes (Dice, Cards). Size is .
  • Continuous: Uncountable outcomes (Choosing a real number between 0 and 1). Size is length, area, or volume.
Example (Two Dice)
Roll two standard 6-sided dice. . . Event .

Practice Problems

Exercise (Problem 1)
Describe the sample space of flipping a coin until a Heads appears, or until 3 Tails have appeared.
Exercise (Problem 2)
Two distinct numbers are selected from . What is the size of the sample space if order matters? What if order doesn't matter?
Exercise (Problem 3)
A stick of length 1 is broken at a random point . Describe the sample space and the event that the piece is longer than 0.7.