1.21.2 Sample SpaceSample 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 Examples of random experiments with their sample spaces are:sample spaces are:

Discrete and continuous sample spacesDiscrete 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 di erent elements, or if the elements can be “listed”, even if an ffinfinite 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 eventsDefinition 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: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 Example: Suppose our random experiment is to pick a person in the class and see what forms of transport they used to get to in the class and see what forms of transport they used to get to campus yesterday.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 AA

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.

SA B

Note:

S SA B A B A

A 、 B OppositeA 、 B disjoint

AB

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.

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.

SBA

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).⊂ ⊂

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=7,8Find the following sets:(1) (2) A∩D;(3)A B ∪(4) (5)

A

C DU

( )A B DI U

2. If the sample space is S=A∪B and if

P(A)=0.8 and P(B)=0.5, find P(A∩B)

3. , , ( ) 0.7, ( ) 0.4,

( ) 0.5 ( )

If A C B C P A P A C

P AB P AB C

，fi nd