Session 1 2014 Version

8/10/2019 Session 1 2014 Version

1/14

Introduction to Probability

RemalynQ. Casem

email: [email protected]

2014 1-1Probability Theory

2014 Probability Theory 1-2

Sample space and events

TOPICS

Axioms of probability

Finite probability spaces

Finite equiprobable spaces

Set and event operations

Properties of probability

Experiment

2014 1-3

Probability Theory

Sample Space and Events

Sample space

Outcome

a procedure that generatesobservable outcomes

any possible observation of anexperiment.

mutually exclusive, collectivelyexhaustive set of all possibleoutcomes.

Sor

i e lement

universal set

finite/discrete

infinite/continuous

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Probability Theory


mutually exclusive

Exampleof two mutually exclusive events:One die is rolled. Event A is rolling a 1 or 2.Event B is rolling a 4 or 5.

Exampleof two non-mutually exclusive events:One die is rolled. Event A is rollinga 1 or 2. Event B is rolling a 2 or 3.

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Probability Theory


collectively exhaustive

Example of collectively exhaustive events:One die is rolled. Event A is rolling anumber less than 5. Event B is rolling anumber greater than 3. Any roll of a die willsatisfy either A or B (in fact, a roll of 4satisfies both).

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Probability Theory


mutually exclusive collect ively exhaustive

Example of events that are mutuallyexclusive and collectively exhaustive:You buy a stock today. The event A is thatthe price of the stock goes up tomorrow.The event B is the price of the stock goesdown tomorrow. Event C is the price ofthe stock does not change.


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Probability Theory


EventE

set of all outcomes (sample points)satisfying some property thatcharacterizes that event.se t

simple

compound

independent

dependent

Experiment

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Probability Theory

Sample Space and Events Example 1

Roll a normal six-sided die once

Each outcome is a number i = 1, , 6

6 distinct numbers: S = {1,2,3,4,5,6}Sample space

Sor

Outcomes

i

Events

E

E1 = set of all odd outcomes

E2 = set of all outcomes greater than 2

E3

= set of all outcomes that are thesquare of an integer

Experiment

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Probability Theory


2 rolls of a four-sided die, recordboth numbers

Pairs of numbers {1,2,3,4} x {1,2,3,4}

16 distinct pairs if order matters;10 distinct pairs if order doesnt matter

Sample space

Sor

Outcomes

i

Events

E

E1

= set of all outcomes witha sum equal to 4

E2 = set of all outcomes with

an odd sum

Experiment

2014 1-10

Probability Theory


2 rolls of a four-sided die, recordthe sum

Sum of the two numbers, a number fro2 and 8

{2,3,4,5,6,7,8}Sample space

Sor

Outcomes

i

Events

E

E1

= set of all even numbers{2, 4, 6, 8}

E2 = set of numbers > 5

{6, 7, 8}

2014 1-11

Probability Theory

Venn Diagrams: Flipping Three Coins

(a)True / FalseAll three Venn diagrams are the samplespace of the outcomes of flipping a cointhree times: {HHH, HHT, HTH, HTT, THH,THT, TTH, TTT}.


TRUE

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Probability Theory


(b) Event A could be described as: flipping a headno times / exactly two times

at least once / at most two times


at least once


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Probability Theory


(c) Event B could be described as: flipping a headno times / exactly two times



at most two times

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Probability Theory


(d) Event C could be described as: flipping a tailno times / exactly two times



at least once

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Probability Theory


(d) Event D could be described as: flipping a headno times / exactly two times



exactly two times


Set Operations

Union = {x/ x in A or x in B}

Intersection = {x/ x in A and x in B}

Sometimes AB is written as A+BAB is written as AB


Complement = {x/ x in S and x not in A}

Difference = {x in A and x not in B}

= (A B = A Bc)

Set Operations


De Morgans Theorems

Set Operations

(1) (A B)c = Ac Bc

NOT in (A or B) = (NOT in A) AND (NOT in B)

(2) (A B)c = Ac Bc


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Event Operations

Let E1 = {a, b, c, d, e, f}E2 = {e, f, g, h}E3 = {i}

S = {a, b, c, d, e, f, g, h, i, j}

(a) E2c = {a, b, c, d, i, j}

(b) E1 E2 = {a, b, c, d, e, f, g, h}

(c) (E1 E2)E3 =

(d) (E1 E2 )c = {i, j}


Concept of Probability

Probability is a measure of the likelihood of anevent taking place once a randomexperiment is conducted.


1) 1 P[A] 0

Axioms of Probability

To every event A E, a real number P[A] thatsatisfies the following three axioms is assigned:

Probability is a nonnegative number.

2) P[S] = 1

S is the sure event.


Axioms of Probability

3) If A B = , then P [A B] = P [A] + P[B]

Additive property for disjoint events


Properties of Probability

i) P[] = 0

Properties derived from the three axioms:

The empty set (null set) is the impossible event.



ii) P [A] + P[Ac] = 1


The empty set (null set) is the impossibleevent.


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iii) P[A B] = P[A] + P [B] P[A B]


Additive property



iv) If A B, then P[B A] = P[B] P[A]and P[A] P[B]


Finite Sample Spaces

Experiments which have finitely

many outcomes are said to havefinite sample spaces.

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Probability Theory

1 2 3 4 1 2 3

If we put these in a hat and pull one out at random, what is:

P(green square) =


7

4

P(numbered 1) =7

2

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Probability Theory

1 2 3 4 1 2 3

If we put these in a hat and pull one out at random, what is:

P(NOT numbered 1) =


7

5

7

5=

7

1

7

2+

7

4P(green OR numbered 1) =

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Probability Theory

Finite Sample Spaces Example

A die is loaded so that the numbers 2, 4 and6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.

(a) One die is rolled. Assign the probabilities tothe outcomes that accurately model thelikelihood of the various numbers to appear.

12

1=P(6)=P(4)=P(2)

12

3=P(5)=P(3)=P(1)

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Probability Theory


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Finite Probability Spaces Example


(b) One die is rolled. What is the probability of getting a 5?

12

3=P(5)

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Probability Theory


A die is loaded so that the numbers 2, 4 and

6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.

(c) One die is rolled. What is the probability of getting an even number?

12

3=P(6)+P(4)+P(2)

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Probability Theory



(d) One die is rolled. What is the probability of not getting a 5?

4

3=12

9=12

3-1

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Probability Theory



(e) Two dice are rolled. What is the probability ofgettingdoubles?

2

2

)12

3(=P(1)P(1)=1)P(1,

)12

1(=P(2)P(2)=2)P(2,

22 )12

33(+)

12

13(=P(doubles)

2014 1-34

Probability Theory


A die is loaded so that the numbers 2, 4 and

6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.

(f) Two dice are rolled. What is the probability ofgetting a sum of 7?

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

)12

312

13(+)

12

112

33(=

)12

112

36(=

2014 1-35

Probability Theory

EquiprobableSpaces

If all the sample points within a given finiteprobability space are equal to each other,then it is known as an equiprobable space.

Examples:

A toss of a fair coin.

Select a name at random from a hat.

Throwing a well balanced die.

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Probability Theory


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EquiprobableSpaces

Theoretical Probability

=> finding the probability of eventsthat come from an equiprobablesample space.

n(S)

n(E)=)(EP

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Probability Theory

EquiprobableSpaces Example 1

A pair of fair dice is tossed. Determine the

probability that

(a) at least one of the dice shows a 6

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

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

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

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

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

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

0.30636

11

2014 1-38

Probability Theory


A pair of fair dice is tossed. Determine theprobability that

(b) the sum of the two numbers is 5

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

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

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

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

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

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

0.1119

1=36

4

2014 1-39

Probability Theory


From a group of 10 women and 5 men, 2people are selected at random to form acommittee. Find the probability that

(a) only men are selected

0.09521

2=

105

10

n(only men selected)= C(5, 2) = 10

n(2 person committees) = C(15, 2) = 105

2014 1-40

Probability Theory


From a group of 10 women and 5 men, 2

people are selected at random to form acommittee. Find the probability of selecting

(b) exactly 1 man and 1 woman

0.47621

10=

105

50

n(exactly 1 man and 1 woman) = C(5, 1) = 50

n(2 person committees) = C(15, 2) = 105

2014 1-41

Probability Theory


One student's name will be picked atrandom to win a CD player. There are 12male seniors, 15 female seniors, 10 male

juniors, 5 female juniors, 2 malesophomores, 4 female sophomores, 11 malefreshmen and 12 female freshman. Find theprobability that

(a) a senior or a junior is picked

0.59271

42=71

15+

71

27

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Probability Theory


8/14


One student's name will be picked atrandom to win a CD player. There are 12male seniors, 15 female seniors, 10 malejuniors, 5 female juniors, 2 malesophomores, 4 female sophomores, 11 malefreshmen and 12 female freshman. Find theprobability that

(b) a freshman or a female is picked

0.66271

47=

71

12-71

36+

71

23

2014 1-43

Probability Theory


One student's name will be picked at

random to win a CD player. There are 12male seniors, 15 female seniors, 10 malejuniors, 5 female juniors, 2 malesophomores, 4 female sophomores, 11 malefreshmen and 12 female freshman. Find theprobability that

(c) a freshman is NOT picked

0.67671

48=

71

23-1=

P(E) = 1 - P(a freshman is picked)

2014 1-44

Probability Theory


Lotto is a game where the player picks 6balls from a possible 49.

(a) What are the odds of getting the jackpot?(The player picks the same 6 balls as the 6chosen at the draw.)

To choose 6 numbers from 49 is C(49,6).Therefore, the odds are

816983,13,

1=

C(49,6)

1

2014 1-45

Probability Theory


Lotto is a game where the player picks 6balls from a possible 49.

(b) What are the odds of getting just 4 balls?

possible winning 4-number combinations= C(6, 4) C(43, 2) = 15 903 = 13, 545

032,1

1=

816983,13,

54513,

2014 1-46

Probability Theory

Techniques of Counting

Mrs. Remalyn Q. Casem

email: [email protected]

2014 1-47Probability Theory


Fundamental Counting Principle and

Tree DiagramsPermutationsCircular PermutationsCombinations

TOPICS


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2014 1-49

Probability Theory

Fundamental Counting Principle

A small community consists of 10 women,each of whom has 3 children. If one womanand one of her children are to be chosen asmother and child of the year, how manydifferent choices are possible?

2014 1-50

Probability Theory


A college planning committee consists of 3freshmen, 4 sophomores, 5 juniors, and 2seniors. A subcommittee of 4, consisting of 1person from each class, is to be chosen. Howmany different subcommittees are possible?

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Probability Theory


How many different 7-place license platesare possible if the first 3 places are to beoccupied by letters and the final 4 bynumbers?

2014 1-52

Probability Theory


How many different 7-place license platesare possible if the first 3 places are to beoccupied by letters and the final 4 bynumbers if repetition among letters ornumbers were prohibited?

2014 1-53

Probability Theory

Tree Diagrams

Determine all the possible outcomes when a

coin is tossed three times.

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Probability Theory

Permutations

How many different ordered arrangements ofthe letters a, b,and c are possible?

abc, acb, bac, bca, cab, cba

Each arrangement is known as a permutation.

The different permutations of the n objects are


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2014 1-55

Probability Theory

Permutations

1. How many different battingorders are possible for a baseballteam consisting of 9 players?

The different permutations of the n objects are

2014 1-56

Probability Theory

PermutationsThe different permutations of the n objects are

2. A class in probability theory consists of6 men and 4 women. An examination isgiven, and the students are rankedaccording to their performance.Assume that no two students obtainthe same score.

(a) How many different rankings arepossible?

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Probability Theory


2. A class in probability theory consists of

6 men and 4 women. An examination isgiven, and the students are rankedaccording to their performance.Assume that no two students obtainthe same score.

(b) If the men are ranked justamong themselves and the women

just among them selves, how manydifferent rankings are possible?

2014 1-58

Probability Theory


3. Ms. Jones has 10 books that she isgoing to put on her bookshelf. Of these, 4 are mathematics books, 3 arechemistry books, 2 are history books,and 1 is a language book. Ms. Joneswants to arrange her books so that allthe books dealing with the samesubject are together on the shelf. Howmany different arrangements arepossible?

2014 1-59

Probability Theory

Permutations

We shall now determine the number of

permutations of a set of n objects whencertain of the objects are the same fromeach other. To set this situation straight inour minds, consider the following example.

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Probability Theory

Permutations

How many different letter arrangements can beformed from the letters PEPPER?

Hence, there are 6!/(3! 2!) = 60 possible letterarrangements of the letters PEPPER.


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2014 1-61

Probability Theory

Permutations

We shall now determine the number of permutations of a set of n objects whencertain of the objects are the same fromeach other. To set this situation straight inour minds, consider the following example.

different permutations of n objects,of which n1 are alike, n2 are alike,... , nr are alike.

2014 1-62

Probability Theory

Permutations

A chess tournament has 10 competitors, of

which 4 are Russian, 3 are from the UnitedStates, 2 are from Great Britain, and 1 isfrom Brazil. If the tournament result lists

just the nationalities of the players in theorder in which they placed, how manyoutcomes are possible?

How many different signals, each consistingof 9 flags hung in a line, can be made from aset of 4 white flags, 3 red flags, and 2 blueflags if all flags of the same color areidentical?

2014 1-63

Probability Theory

Combinations

To determine the number of different groupsof r objects that could be formed from atotal of n objects

A committee of 3 is to be formed from agroup of 20 people. How many differentcommittees are possible?

2014 1-64

Probability Theory

Combinations

From a group of 5 women and 7 men, howmany different committees consisting of 2women and 3 men can be formed?

What if 2 of the men are feuding and refuseto serve on the committee together?

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory


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EXERCISES

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory


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EXERCISES

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Probability Theory

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory


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EXERCISES

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Probability Theory

EXERCISES

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Probability Theory

EXERCISES

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Probability Theory


End of Presentation

Documents

Session 1 2014 Version