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Probability Basics: Sample Spaces and Events

Chapter 2: Probability

This chapter introduces probability from a mathematical perspective, moving beyond everyday notions of chance.

Section 1: Sample Spaces and Events

Experiment

An experiment is any action or process with an uncertain outcome. It is subject to some uncertainty.

Examples:

  • Tossing a coin: The outcome is either heads or tails, but the result is uncertain before the toss.

  • Throwing a dice: A six-sided die results in an outcome from 1 to 6, with uncertainty about which number will appear.

Sample Space

A sample space, denoted by S, is the set of all possible outcomes of an experiment. S is the set of outcomes of an experiment.

Examples:

  • Tossing a coin: The sample space is S = {Head, Tail}.

  • Throwing a dice: The sample space is S = {1, 2, 3, 4, 5, 6}.

  • Two Gas Stations: Consider two gas stations at an intersection, each with six pumps. An ordered pair (x, y) represents the number of pumps in use at Station 1 (x) and Station 2 (y). For instance, (3, 5) means three pumps are in use at Station 1 and five at Station 2. The set of all elements in the sample space consists of 36 elements.

Sample Space with Infinite Number of Elements

Consider a battery testing scenario. If a battery's voltage is within prescribed limits, it's a success; otherwise, it's a failure. The experiment involves testing batteries one by one until a successful battery is found. The sample space consists of outcomes like {Success, Failure Success, Failure Failure Success, …}, representing the sequence of tests until the first success. This gives rise to an infinite number of elements in the set.

Event

An event is a subset of the outcomes in a sample space S. Events can be simple or compound, depending on the number of elements in the subset. A subset will actually be the set consisting of elements fewer in number than the original subset.

Example: Three Vehicles at a Freeway Exit

Consider an experiment where three vehicles take a freeway exit and turn either left (L) or right (R) at the end of the exit ramp. The sample space consists of eight possible outcomes: {LLL, LLR, LRL, LRR, RLL, RLR,RRL, RRR}.

  • Event A: Exactly one of the three vehicles turns right. This event consists of the subset {LLR, LRL, RLL}.

  • Event B: At most one of the vehicles turns right. This event includes {LLL, LLR, LRL, RLL}.

  • Event C: All three vehicles turn in the same direction. This event consists of {LLL, RRR}.

Events A, B, and C are compound events because they contain multiple elements.

Operations on Sets: Union, Intersection, Complement

Venn diagrams are helpful for visualizing relationships between sets.

  • Intersection: The intersection of sets A and B includes elements that are in both A and B.

  • Complement: The complement of set A includes all elements in the sample space S that are not in A.

Examples:

  • Let A = {0, 1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {1, 3, 5}.

    • The union of A and B, denoted as A \cup B, is {0, 1, 2, 3, 4, 5, 6}.

    • The union of A and C, denoted as A \cup C, is {0, 1, 2, 3, 4, 5}.

    • The intersection of A and B, denoted as A \cap B, is {3, 4}.

    • The intersection of A and C, denoted as A \cap C, is {1, 3}.

    • If the sample space S = {0, 1, 2, 3, 4, 5, 6}, then the complement of A, denoted as A', is {5, 6}.

Disjoint or Mutually Exclusive Sets

If two sets, A and C, have nothing in common (i.e., their intersection is empty), they are said to be disjoint or mutually exclusive.

Generalization to Multiple Sets

The concepts of union, intersection, and complement can be generalized to more than two sets. For instance, one can consider the union or intersection of three sets (A, B, and C) or even an infinite number of sets.