PROBABILITY

Introduction

Every trial or experiment has one or more possible outcomes

An event, denoted by capital letter is a collection of one or more of those outcomes.

For example, in tossing a die, event may be defined as getting odd numbers which consist of three possible outcomes.

The probability denoted by of an event A is a measure of the likelihood or chance of the event occurring in any one trial or experiment.

The value for P(A) is between 0 and 1. P(A)=0 mean there are no chance for an

event to occur (impossible event) P(A)=1 means an event that is sure to

occur (certain event).

Venns Diagram

Sample spaces, S contains all possible outcomes of an experiment.

Event is a subset of the sample space. For example, event A is a subset of S, .

A B

S

C

The union of event A and B, which is denoted as is the event of all elements that belong to A or B or both.

The intersection of event A and B, which is denoted by , is the event of all elements that belong to both A and B.

Event A and C have no outcomes in common, they are said to be mutually exclusive.

A B

A B

Classical definition of probability

If S is a sample space, event A is a subset of S, and the experiment has an equal probability of accurance, then the probability of A can be obtain by:

Number outcome in A ( )( )

Number outcome in S ( )

n AP A

n S

Example 1Karim has a set of eight cards numbered 1 to 8. A card is drawn randomly from the set of cards. Find the probability that the number drawn is

(a) 8(b) Not 8

Solution:(a) If E is the event that a number 8 is

drawn and S is the sample space, then S={1,2,3,4,5,6,7,8} ; E={8} ; ={1,2,3,4,5,6,7}

(b)

( ) 1( )

( ) 8

n EP E

n S

( ) 7( )

( ) 8

1 7Note that 1 ( ) 1 ( )

8 8

n EP E

n S

P E P E

E

Example 2Two fair dice are thrown. Determine I. the sample space of the experiment.II. the elements of event A if the outcomes of

both dice thrown are showing the same number of dot.

III. the elements of event B if the first thrown giving a greater number of dot than the second thrown.

IV. probability of event A, P(A) and event B, P(B)

If A and B are two events from an experiment with the conditions that

Where is the event that A occurs or B occurs or both events A and B occurs while is the event that both A and B occur together

( ) 0 and ( ) 0, thenP A P B

P A B P A P B P A B

A BA B

Let say n(A)=a, n(B)=b, n(S)=n and n(A B)=d

S A B

a-d b-dd

n A BP A B

n S

a d d b d

na b d

na b d

n n nP A P B P A B

Example 3In a class consisting of 25 students, 10 are girls and15 are boys. 4 of the girls and 8 of the boys wear glasses. If a student is selected at random, what is the probability that the student selected is a girl or wearing a glasses?

SolutionIf A is event that selected student is a girl and B is event that the selected student wear glasses.

10 12

25 25

n A n BP A P B

n S n S

425

10 12 425 25 251825

n A BP A B

n S

P A B P A P B P A B

18Probability that the student selected is a girl or wearing glasses is

25

If event A can occurs or even B can occurs but not both event A and B can occurs, then two even A and B are said to be mutually exclusive.

This mean that if A and B are mutually exclusive event, then if A occurs, B cannot occur and if B occurs, A cannot occur.

For two mutually exclusive event A and B,

Probability Of Mutually Exclusive Event

0 and

A B

P A B

P A B P A P B

S

Probability of two mutually exclusive event

A B

ExampleFive graduates of equal ability apply for vacancy. Only one applicant will be successful. The applicant are Ali, Bakri, Chandran, David and Eng Kok. Find the probability that

(a) Bakri will be successful(b) Bakri or david will be successful

Solution(a)Each applicant has an equal chance of

getting the job and selections are mutually exclusive because only one will be chosen.

(b) P(Bakri or David)

1 1 25 5 5

P Bakri P David

Ali, Bakri, Chandran, David, Eng Kok , 5

1Therefore, Bakri 0.2

5

S n S

P

Probability If Three Events

The result of probability for two event

can be extended to three event as follows:If A, B and C are three events from sample

space S, then,

P A B P A P B P A B

P A B C P A P B P C P A B P A C

P B C P A B C

Usually, problems involving probability of three events can be solved with the aid of Venn diagram.

A B

C

S

A B C

A B

A C B C

P A B C P A P B P C P A B P A C

P B C P A B C

Tree Diagram

When a trial or an experiment of interest consist of a sequence of several stages, it is convenient to represent these with tree diagram.

A tree diagram highlights all the possible outcomes and systematically works out the corresponding probabilities.

Example 2.6 A coin is tossed three times. If A is the event

that a head occurs on each of the first two tossed, B is the event that a tail occurs on the third toss and C is the event that exactly two tails occur in the three tosses. Show a tree diagram for this experiment and list the possible outcomes of the sample space, S and events A, B and C. Find the probability for events A, B and C to occur.

Answer for exercise

1. Let A be a event that a student like to read megazine AA and B the event that a student like to read megazine BB.

P(A)=0.8

Put these information into Venn diagram.

0.1P A B 0.05P A B

0.70 0.10 0.15

0.05

S A B

a) Probability that the student selected like to read megazine BB is 0.25

b) P(student like to read AA or BB but not both)=0.70+0.15=0.85

0.15 0.10

0.25

P B

2.(a)

(b)

0.65 0.4 0.15

0.65 0.4 0.15

0.4

P E M P E P M P E M

P M

P M

0.8 0.4 0.2

0.6

P E S P E P S P E S

P S

P S

P E M S P E P M P S P E M

P E S P M S

P E M S

(c)

0.95 0.4 0.4 0.6 0.15 0.2 0.2

0.1

P E M S

P E M S

not a member of any society

1

1 0.95

0.05

P

P E M S

3.The sample space S={3 black balls, 4

white balls, 9 yellow balls} n(S)=16

Let A: event that black balls is chosen B: event ball is chosen

(a)

(b)

316

n AP A

n S

because A and B are mutually exclusive

3 9 316 16 4

P A B P A P B

n A n B

n S n S

4(a) Let A:Event that Aru be a champion

B:Event that Bakri be a champion C:Event that Chan be a championSince only one person will be a champion, event A,B and C are mutually exclusive. Thus

Probability that one of them will be a champion is 0.6

0.2 0.3 0.1

0.6

P A B C P A P B P C

(b)

Probability that none of them will be a champion is 0.4

1

1 0.6

0.4

P A B C P A B C