Chapter 2 : Probability

Definitions

The term probability refers to the study of randomness and uncertainty.

An experiment is any activity or process whose outcome is subject to uncertainty.

The sample space of an experiment, denoted by \(\mathscr{S}\), is the set of all possible outcomes of that experiment.

An event is any collection (subset) of outcomes contained in the sample space \(\mathscr{S}\). An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome.

  1. The complement of an event A, denoted by \(A'\), is the set of all outcomes in \(\mathscr{S}\) that are not contained in A.
  2. The union of two events A and B, dentoed by \(A\cup B\) and read “A or B”, is the event consisting of all outcomes that are either in A or in B or in both events (so that the union includes outcomes for which both A and B occur as well as outcomes for which exactly one occurs) - that is, all outcomes in at least one of the events.
  3. The intersection of two events A and B, denoted by \(A\cap B\) and read “A and B”, is the event consisting of all outcomes that are in both A and B.

Let \(\emptyset\) denote the null event (the event consisting of no outcomes whatsoever). When \(A\cap B=\emptyset\), A and B are said to be mutually exclusive or disjoing events.

Theorems and Axioms

Axiom 1. : For any event A, \(P(A)\geq 0\).

  • the chance of A occuring should be nonnegative

Axiom 2. : \(P(\mathscr{S} =1)\).

  • \(\mathscr{S}\) contains all possible outcomes

Axiom 3. : If \(A_1\), \(A_2\), \(A_3\), … is an infinite collection of disjoint events, then \[P(A_1\cup A_2\cup_2 A_3\cup ...)=\sum_{i=1}^{\infty} P(A_i)\]

  • the chance that least one event will occur, and no two events can occur simultaneously, is the sum of the chances of the individual events.

Proposition : \(P(\emptyset )=0\) where \(\emptyset\) is the null evetn (the event containing no outcomes whatsoever). This in turn implies that the property contained in Axiom 3 is valid for a finite collection of dijoint events.

Proposition : For any event A, \(P(A)+P(A')=1\), from which \(P(A)=1-P(A')\)

Proposition : For any event A, \(P(A)\leq 1\).

Proposition : For any two events A and B, \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]