1.2 Sample Space

Definition 1.1 The set of all possible outcomes of a random experiment is called the sample space of the experiment and is denoted by S or Ω. The possible outcomes themselves are called sample points or elements and are denoted by e1,e2 etc.

Examples of random experiments with their sample spaces are:

Discrete and continuous sample spaces Definition: A sample space is finite if it has a finite number of elements. Definition: A sample space is discrete if there are “gaps” between the different elements, or if the elements can be “listed”, even if an infinite list (eg. 1, 2, 3, . . .). In mathematical language, a sample space is discrete if it is countable. Definition: A sample space is continuous if there are no gaps between the elements, so the elements cannot be listed (eg. the interval [0, 1]).

Definition of events Definition 1.2: An event is a subset of the sample space. That is, any collection of outcomes forms an event. Events will be denoted by capital letters A,B,C,.... Note: We say that event A occurs if the outcome of the experiment is one of the elements in A. Note: Ω is a subset of itself, so Ω is an event, it is called certain event. The empty set, ∅ = {}, is also a subset of Ω. This is called the null event, or the event with no outcomes.

Examples of events are: 1. The event that the sum of two dice is 10 or more, 2. The event that a machine lives less than 1000 days, 3. The event that out of fifty selected people, five are left-handed,

Example: Suppose our random experiment is to pick a person in the class and see what forms of transport they used to get to campus yesterday.

Opposites: the complement or ‘not’ operator Definition 1.3 The complementof an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement of A by the symbol of . S B A

2.the intersection ‘and’ operator Definition 1.4 The intersection of two events A and B, denoted by the symbol A∩B or A, is the event containing all elements that are common to A and B.

Definition 1.5 Two events A and B are said to be disjointif A∩B =Ø. More generally the events A1,A2,A3,…… are said to be pairwise disjoint or mutually exclusive if Ai ∩ Aj=Ø whenever i≠j. S A B

Note: S A B A、B Opposite A、B disjoint disjoint Opposite A∪B=S and AB=Ø

the union ‘or’ operator Definition 1.6 The union of the two events A and B, denoted by the symbol A∪B, is the event containing all the elements that belong to A or B or both.

An event A is said to imply an event B if A⊂B. Definition 1.7 An event A is said to imply an event B if A⊂B. This means that if A occurs then B necessarily occurs since the outcomes of the experiment is also an element of B. S B A

Definition 1.8 An event A is equal to event B if and only if (iff) A⊂B and B⊂A (denoted by A=B).

Experiment: Pick a person in this class at random. Examples: Experiment: Pick a person in this class at random. Sample space: Ω = {all people in class}. Let event A =“person is male” and event B = “person travelled by bike today”. Suppose I pick a male who did not travel by bike. Say whether the following events have occurred:

A =“person is male” B = “person travelled by bike today”. pick a male who did not travel by bike

Properties of union, intersection, and complement

Distributive laws

Exercise 1. Consider the set S={1,2,3,4,5,6,7,8,9} with subsets A={1,3,5,7,9},B={2,4,6,8},C={1,2,3,4}, D={7,8} Find the following sets: (1) (2) A∩D;(3)A∪B (4) (5)

2. If the sample space is S=A∪B and if P(A)=0. 8 and P(B)=0 2. If the sample space is S=A∪B and if P(A)=0.8 and P(B)=0.5, find P(A∩B)