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STAT 111

Chapter TwoProbability

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Probability

Many statistical principles and procedures are basedon the important concept of probability. The purpose ofthis chapter is to define such concept and discuss some of its

properties. A random experiment is an experiment in which1. All possible outcomes of the experiment are known in

advance,

2. Any performance of the experiment results in an outcomethat is not known in advance,

3. The experiment can be repeated under identical conditions

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Examples

1) Tossing a coin once or several times.

2) Obtaining blood types from a group of

individuals.

3) Determining the sex of a newborn.

4) Tossing two dice.

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Definition

Definition: The sample space of an experiment, denoted byS, is the set of all possible outcomes of that experiment.

Sample spaces are either1. Discrete : which contains a finite number of elements , or

and infinite but countable number of elements .

2. Continuous : which contains an infinite number of sample

points constituting a continuum , such as all points on a linesegment or all the points in a plane .

S

Discrete Continuous

(infinite)Infinite

(Countable)Finite

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Example

Describe an appropriate sample space for the experiments

below. Determine the number of elements and state whether

the sample space is discrete or not.

A coin is tossed two times.

H

H

H

T

T

TS = {HH, HT, TH, TT} n(S) = 4, discrete

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Example (continued)

A coin is tossed and a dye is rolled

S = (H,T)x(1,2, ...,6)

S = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)},

n(s) = 2x6=12, discrete

Two numbers are selected from the set {1, 2, 3} without repetition ofdigits.

S = {(1,2), (1,3), (2,1), (2,3), (3,1), (3,3) }

n(S) = 3x2=6, discrete(1,2)

32

3

2

1

1

(1,3)

(3,2)

(2,1) (3,1)

(2,3)

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A coin is tossed until the first head appears.

S = {H, TH, TTH, ...,), and so the coin is tossed an infinitenumber of times, here , refers to the case when a head

never appears.

A light bulb is observed so that the length of its useful lifemight be recorded.

S = {(0, )}, since one could not say with certainty that thebulb would have burned out by any given time => S iscontinuous.

Example (continued)

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Events

Definition An event is any subset of the sample space S. AneventA occurs, if the outcome of the experiment is in A.

Example Two cards are drawn, randomly with replacement fromthree cards carrying the number 1,3, 5. Describe the samplespace, find the events

A= {sum of the two numbers is 5 }

B= {sum of the numbers is at least 6}A=

B={(1,5), (3, 3), (3, 5), (5,1), (5, 3), (5, 5)}

(1,3)

53

5

3

1

1

(1,5)

(5,3)

(3,1) (5,1)

(3,5)

(3,3)

(5,5)

(1,1)

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Example

Example:

An experiment consists of rolling a die until a 3 appears,

1) Construct a sample space for this experiment. Let 3 denotes 3appears , let A denote 3 does not appear

S = { 3 , A 3 , AA3 , } .

2) List the elements in E , the event that 3 appears before the fifthroll

E = { 3 , A3 , AA3 , AAA3 }

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Some Relations from Set Theory

An event is nothing but a set, so that relationships andresults from elementary set theory can be used to studyevents. The following concepts from set theory will beused to construct new events from given events.

1) The union of two events A and B denoted by A B andread 'A or B' is the event consisting of all outcomes whichare either in A or in B or in both events, that is

A U B= {x S :x A or x B}

If A1, A2, ... are events, the union of these events, denoted, byis defined to be that event which consists of all points that

are in An for at least one value of n = 1,2, ...

1nnA

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Some Relations from Set Theory

2) The intersection of two events A andB, denoted by A B and read 'Aand B' is the event consisting of all outcomes which are in both A and B,that is,

A B={x S :x A and x B}

If A1, A2, ... are events, the intersection of these events, denoted by , isdefined to be that event consisting of all points that are in all of

An, n= 1 , 2 ,

3) The complement of an event A , denoted by A ( or Ac ) is the set of alloutcomes in S which are not contained in A . That is

Ac = { x S : x A }

1n

nA

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Some Relations from Set Theory

Definition

When A and B have no outcomes in common (the

intersection of A and B contains no outcomes,

i.e. A B= ), they are said to be mutually exclusive of

disjoint events.

DefinitionA nonempty collection of subsets F of a set S is

called a -filed of subsets of S provided the following

properties hold.

FinbothareAandAthen1,2,...,nF,AIf3.

FAthenFAIf2.

FS.1

1n

n

1n

nn

c

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Probability

Definition

A probability measure P (or simply probability) is a real-valued functionhaving domain F satisfying the following properties.

Axioms of Probability:

1- 0 P(A) 1 F

2- P(S)=1

3- If An, n = 1,2, ..., are mutually disjoint sets in S, then

A probability space , denoted by ( S , F , P ) is a field of subsets F and aprobability measure P defined on S .

11 n

n

n

nAAP

A

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Probability

Definition

An assignment of probability is said to be equally likely (or

uniform) if each elementary event in S is assigned the

same probability.

Thus if S contains n elements wi, i.e. if S = { w1, ,wn}

then

Sn

An

Sinelementsofnumber

AinelementsofnumberAP

assignmentwith this

n

1

wP i

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Properties of probability

In the following, some additional properties of probability measure P willbe presented. These properties follow from the definition of a probabilitymeasure.

1) P( ) = 0

2) Let A1, A2, ..., An a collection of pairwise disjoint events in S, then

3) If A is an event in S, then P(A)=1-P(A)

4) Let A and B be events in S , then P (A ) = P (A B ) + P ( A Bc)

5) Let A and B be event in S, then P(A U B ) = P(A)+ P(B) - P(A B)6) P(A U B U C) = P(A) + P(B) + P(C) - P(A B) - P(A C) - P(B C) +

P(A BC)

7) If A B , then P ( A ) ( B )

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Example 1

Example 1:

If two dice are rolled what is the probabilitythat

1) The sum of upturned faces will equal7?

2) The sum of upturned faces will equal2 or 12?

3) The sum of upturned faces will be aneven number or a number less than6? Let A = {sum is even), B ={number less than 6}

6 (1,6) (2,6) (3, 6) (4, 6) (5,6) (6, 6)

5 (1,5) (2,5) (3, 5) (4,5) (5, 5) (6,5)

4 (1,4) (2,4) (3, 4) (4, 4) (5, 4) (6, 4)

3 (1,3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3)

2 (1,2) (2,2) (3, 2) (4, 2) (5, 2) (6, 2)

1 (1,1) (2, 1) (3,1) (4,1) (5, 1) (6,1)

1 2 3 4 5 6

61

36

67sum P

6

2

36

1

36

112sum2sum PP

36

24

36

4

36

10

36

18BABABA PPPP

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Example 2

Given P(A) = 0.59, P(B) = 0.30, P (A B) = 0.21,

find

1) P ( A U B ) = 0.59 + 0.30 0.21 = 0.68

2) P (A Bc) = P ( A )P ( A B ) = 0.59 - 0.21 = 0.383) P ( Ac U Bc ) = 1- P ( A B ) = 1 0.21 = 0.79 or

P (AcUBc)=P(Ac)+P(Bc)P(AcBc)=0.41 +0.7 -0.32=0.79

4) P ( Ac Bc) = 1- P (A U B ) = 1-0.68 =0.32

A Ac Total

B 0.21 0.09 0.3

Bc 0.38 0.32 0.7

Total 0.59 0.41 1

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Example 3

If A and B are two events

such that A B. What is

P(AB), what is P(A

B) and what is P(A Bc)?

P(AB)=P(B)

P(A B)=P(A)

P(A

B

c

)=P(A)- P(A

B)= P(A)- P(A)=0

S

BA

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Example 4If A, B and C are mutually exclusive events and P(A) = 0.2,

P(B) = 0.3, and P(C) = 0.2, find

1. P(A U B U C)?

P(A U B U C)= P(A) + P(B) + P(C) = 0.2 + 0.3 + 0.2 = 0.7

2. P(Ac (B U C))?

P(Ac

(B U C))=P(B) + P(C) = 0.3 + 0.2 = 0.5

0.3

0.2 0.2

0.3

A

B

C S

A

B

C S

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Example 5

A two-digit number is formed by randomly selecting, with replacement,digits from the set {6, 7, 8, 9}. Find the probability;

1. The two digits are the same?

n(S)=4x4=16n(two digits are the same)=4x1=4

P(two digits are the same)=4/16=1/4

2. The number is odd?n(number is odd)=4x2=8

P(number is odd)=8/16=1/2

Remember

SnAn

AP

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Example 6

If 10 people, including A and B, are randomly arranged in a

line,

1. What is the probability that A and B are next to eachother?

2. What would probability be if the people were randomlyarranged in a circle?

!10

)!110(!2

Remember

SnAn

AP

!10

)!210(!2

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Independent Events

Suppose that two events A and B occur independently of one another in thesense that the occurrence or nonoccurrence of either of them has norelation to the occurrence or nonoccurrence of the other, that is theprobability that A and B will occur is equal to the product of their individualprobabilities.

Definition

Two events A and B independent if and only if

P ( A B ) = P ( A ) P ( B )

More generally , for n 3 , events A1 , A2, , An are independent if

P ( A1 A2 An) = P (A1) P(A2)P(A3) P ( An)

And if any subcollection containing at least two but fewer than n eventsare independent .

Events that are not independent are said to be dependent. Note thatindependence of events is not to be confused with disjoint or mutuallyexclusive events.

ResultAssuming that A and B are independent events, then the events Acand B, A and Bc, Ac and Bc are also independent.

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Example 1

Let two fair coins be tossed, let A = {head on the second

throw}, B = {head on the first throw}. Show that A and B

are independent.

S = {HH, HT, TH, TT}

A = {HH, TH}

B = {HH, HT}

AB={HH}P(AB)=1/4

P(A)P(B)= x =1/4= P(AB)

A and B are independent

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Example 2

Two students A and B are both registered for a certain course.

If student A attends class 80 percent of the time and student B attendsclass 60 percent of the time, and if the absence of the two studentsare independent, what is the probability that at least one of the two

students will be in class on a given day?

P(A) = 0.8, P(B) = 0.6

Ac and Bc are independent A and B are also independent

P ( A U B ) = P ( A ) + P (B) P (A ) P (B ) = 0.8 + 0.6 0.6x0.8=0.92

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Example 3

A coin is biased so that a head is twice as likely to occur as a tail, If the coin is tossed 3times, what is the probability of getting 2 tails and 1 head?

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT},

let P(H) = 2W, P(T) = 1W

2W + 1W =1 W=1/3P(H) = 2/3 and P(T) = 1/3

A = {TTH,THT,HTT},

P(TTH) = P(T) P(T) P(H)= 1/3 x 1/3 x 2/3 = 2/27 by independence

P(THT) = P(T) P(H) P(T)= 1/3 x 2/3 x 1/3 = 2/27 by independence

P(HTT) = P(H) P(T) P(T)= 2/3 x 1/3 x 1/3 = 2/27 by independence

P(A) = P(TTH) + P(THT) + P(HTT)= 2/27 + 2/27 + 2/27=6/27=2/9

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Example 4

Suppose that three balanced coins are tossed. Find the probability that at least one of

them lands head.

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT},

A1=first coin lands head= {HHH, HHT, HTH, HTT}

A2=second coin lands head={HHH,HHT,THH,THT} andA3=third coin lands head={HHH, HTH, THH, TTH}

P(A1)=4/8=1/2

P(A2)=4/8=1/2

P(A3)=4/8=1/2

P ( at least one lands head ) = P (A1 U A2 U A3)

8

7

8

11

2

1

2

1

2

111

1

321

321321321

ccc

ccccccc

APAPAP

AAAPAAAPAAAP

APAP c 1

Remember

laws)sMorgan'(DeBUA)B(Accc

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Example 5A system containing five components is arranged in the manner shown in

Figure - 1, where the probabilities given indicate the chance that thecomponent will work. If we assume that whether a component works or notis independent of whether any other component is working or not. what isthe probability that the system will work?

P(system work) =P(A (B U C) (D U E))= P(A)P(B U C)P(D U E) since independent

P(B U C)=1- P(Bc Cc)= 1- P(Bc)P( Cc)=0.995 since independent

Or ( B U C ) = P ( B ) + P ( C ) P (B ) P (C) =0.995 since independent

Similarly, P(D U E)= 1- P(Dc Ec)= 1- P(Dc)P( Ec)=0.9979

P(system work) = P(A)P(B U C)P(D U E)=0.98x0.995x0.9979=0.973

P(E)=0.97

P(D)=0.93

P(C)=0.95

P(B)=0.90

P(A)=0.98