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Chapter 5 Chapter 5

Elementary Probability Elementary Probability

5.1 Introduction 5.1 Introduction

Probability is:



A quantitative measure of uncertainty



A measure of the strength of belief in the

occurrence of an uncertain event



A measure of the degree of chance or likelihood of

occurrence of an uncertain event



Measured by a number between 0 and 1 (or

between 0% and 100%)

Complement of a Set

A

A

S

Venn Diagram illustrating the Complement of an event

Venn Diagram illustrating the Complement of an event



Intersection (And)

• a set containing all elements in both A and B



Union (Or)

• a set containing all elements in A or B or both

 Mutually exclusive or disjoint sets

• sets having no elements in common, having no

intersection, whose intersection is empty set

 A^  B   A^  B 

 A^  B   A^  B 

Sets: A Union B

A A  BB

A

B

S

Mutually Exclusive or Disjoint Sets

A

B

S

Sets have nothing in common



Sample Space or Event Set

 Set of all possible outcomes (universal set) for a given

experiment

 E.g.: Roll a regular six-sided die

 (^) S = {1,2,3,4,5,6}



Event

 Collection of outcomes having a common characteristic

 E.g.: Even number

 (^) A = {2,4,6}

 Event A occurs if an outcome in the set A occurs



Probability of an event

 Sum of the probabilities of the outcomes of which it consists

 P(A) = P(2) + P(4) + P(6)

Equally-likely outcomes

• For example:^ Throw a die
  • Six possible outcomes {1,2,3,4,5,6}
  • If each is equally-likely, the probability of each is 1/6 = 0.1667 =

16.67%



• Probability of each equally-likely outcome is 1 divided by the number

of possible outcomes

 (^) Event A (even number)

• P(A) = P(2) + P(4) + P(6) = 1/6 + 1/6 + 1/6 = 1/
• for e in A

P A P e

n A

n S

( ) ( )

( )

( )



  



3

6

1

2

P e

n S

( )

( )



1

 To assign probabilities for an event, the possible outcomes of a

random experiment should be counted. The following principles

helps to determine the number of possible outcomes favoring a

given event.

1. Addition Rule

 If a task can be accomplished by k distinct procedures where the

ith^ procedure has ni alternatives, then the total number of ways of

accomplishing the task equals

n 1 + n 2 +…+nk

 Example: Suppose that a man wants to make a journey from

Addis Ababa to Djibouti. The following are the means of

transportation. Air transport: 2 flights; Vehicles: 4 alternatives;

Train: 2 alternatives. (The total alternatives are 2+4+2= 8 )

5.4 Counting rules 5.4 Counting rules

2. Multiplication Principle

 If a choice consists of k steps of which the 1st^ can be made in n 1

ways, the 2nd^ can be made in n 2 ways,…, and the kth^ can be made

in nk ways, then the whole choice can be made in

n 1 .n 2 ….nk ways

 Example: If a test consists of 10 multiple choice questions,

with each permitting 4 possible answers, how many ways are

there in which a student gives his/her answers?

4x4x4x…x4=4^10 ways

= 1, 048, 576 ways of completing the exam.

 Example 5.8: Suppose that we have four letters a, b, c, d.

 What is the number of possible arrangements of these letters taken all at a

time?

4! = 432*1 = 24

 (^) What is the number of possible arrangements of these letters if we use

only three of the letters at a time?

4 P 3 = 24

4. Combination

 It is the possible selections of r items from a group of n items

regardless of the order of selection. The number of

combinations is denoted and is read as n choose r (nCr).

 The number of combinations of r out of n elements is:

 Order is not important

r n r

n

C

r

n

n r 

 Range of Values for P(A):

 Sample space (S): P(S) = 1

 Complements - Probability of not A

 Intersection - Probability of both A and B

 Mutually exclusive events (A and C) :

 Range of Values for P(A):

 Sample space (S): P(S) = 1

 Complements - Probability of not A

 Intersection - Probability of both A and B

 Mutually exclusive events (A and C) :

0  P ( A ) 1

P ( A )  1  P ( A )

P A B

n A B

n S

P ( A  C ) 0

5.5 Probability of an event 5.5 Probability of an event

The Axioms of Probability

• Union - Probability of A or B or both (rule of unions)

Mutually exclusive events: If A and B are mutually exclusive, then

• Union - Probability of A or B or both (rule of unions)

Mutually exclusive events: If A and B are mutually exclusive, then

P A B

n A B

n S

( ) P A P B P A B

( )

( )

  ( ) ( ) ( )

    

P ( A  B ) 0 so P ( A  B ) P ( A ) P ( B )