Probability Basics (1)

7/27/2019 Probability Basics (1)

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Probability and Statistics

Probability


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Define Experiment, Outcome, Simple event, Event, SampleSpace,

Operations and Properties of Events: Union, Intersection, Complement,

Mutually exclusive

3. Explain How to Assign Probabilities

Learning Objectives


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Experiments, Outcomes, Events &

Sampe Space


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An experimentis the processby which an observation (ormeasurement) is obtained.

Experiment: Record an age Experiment: Toss a die

Experiment: Record an opinion (yes, no)

Experiment: Toss two coins

Basic Concepts


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A simple eventis the outcomethat is observed on asingle repetition of the experiment. The basic element to which probability is applied. One and only one simple event can occur when the

experiment is performed.

A simple eventis denoted by E with a subscript.

Basic Concepts


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Each simple event will be assigned a probability, measuringhow often it occurs.

The setof all simple events of an experiment is called thesample space, S.

Basic Concepts


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Experiment

Process of Obtaining an Observation, Outcome or Simple Event

Simple eventMost Basic Outcome of an Experiment

Sample Space

Collection of AllPossible Outcomes

Experiments & Outcomes


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The die toss:

Simple events: Sample space:

Example

1

2

3

4

5

6

E1

E2

E3

E4

E5

E6

S ={E1, E

2, E

3, E

4, E

5, E

6}

SE1

E6E2

E3

E4

E5


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1. Select 1 Card, Note Color

{ Red, Black }2. Play a Football Game

{Win, Lose, Tie}

Example

Experiment Sample Space


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1. Toss a Coin, Note Face

{ Head, Tail}

2. Toss 2 Coins, Note Faces

{HH, HT, TH, TT}

3. Select 1 Card, Note Kind{ 2, 2, ..., A(52)}

4. Select 1 Card, Note suit

{ , , , }

Example

Experiment Sample Space


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An (compount) event is a collection of one or more simpleevents.

Basic Concepts

The die toss: A: an odd number B: a number > 2

S

A ={E1, E3, E5}

B ={E3, E4, E5,E6}

BA

E1

E6E2

E3

E4

E5


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Events

An eventis any collection (subset) ofoutcomes contained in the sample space S.An event is simpleif it consists of exactly oneoutcome and compoundif it consists of more

than one outcome.


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1. Sample Space {HH, HT, TH, TT}

2. 1 Head & 1 Tail {HT, TH}

3. Heads on 1st Coin {HH, HT}4. At Least 1 Head {HH, HT, TH}

5. Heads on Both {HH}

Event Examples

Experiment: Toss 2 Coins. Note Faces.

Event Outcomes in Event


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1. Sample Space , , ,

2. subet 1 ,

3. subset 2 , 4. subset 3 , , ,

5. subset 4 , , ,

Event Examples

Experiment: Select 1 Card, Note suit.

Event Outcomes in Event


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Sample Space


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S

HH

TT

THHT

Sample SpaceS = {HH, HT, TH, TT}

Venn Diagram

Outcome


Event


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2nd Coin

1st Coin Head Tail Total

Head HH HT HH, HT

Tail TH TT TH, TTTotal HH, TH HT, TT S

Contingency Table


S = {HH, HT, TH, TT} Sample Space

OutcomeEvent(Head on

1st Coin)


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Tree Diagram

Outcome

S = {HH, HT, TH, TT} Sample Space


T

H

T

H

T

HH

HT

TH

TT

H


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Compound Events


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1. Intersection

Outcomes in Both Events A andB AND Statement

Symbol (i.e., A B)

2. Union

Outcomes in Either Events A orB or Both

OR Statement

Symbol (i.e., A B)

3. Complement

FormingCompound Events


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Two events are mutually exclusive if, when one event occurs,the other cannot, and vice versa.

Two events A and B are mutually exclusive if = wheredenote empty set.

Event Relations

Experiment: Toss a die A: observe an odd number

B: observe a number greater than 2 C: observe a 6 D: observe a 3

B and C?B and D?


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S

A B

The intersectionof two events, Aand B, is the

event that both AandB occur when theexperiment is performed. We write A B.

Event Relations

A B

If two events A and B are mutuallyexclusive, then P(A B) = 0.


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S

The unionof two events, A and B, is the event that either A orBor bothoccur when the experiment is performed. We write

Event Relations

A BA B


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S

Thecomplementof an event A consists of alloutcomes of the experiment that do not resultin event A. We write AC.

Event Relations

A

AC


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Rolling a die. S = {1, 2, 3, 4, 5, 6}. LetA= {1, 2, 3}

andB= {1, 3, 5}. Find , , , .

Solution:

= 1, 3 ,

= 1, 2, 3, 5 ,

= 4,5,6 , = {2,4,6}

Example


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Deck of Cards


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A deck of cards has 52 cards without counting jokers;

It consist 10 number cards(A[1],2 3,4 5,6 7,8,9,10) and 3

face cards(Jack, Queen, King). There are 4 suits(Club, Spade, Heart , Diamond ).

There are 4 of each number and 13 of each suit.

Deck of Cards


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S

Black

Complement of Event Example

Event Black:

2B, 2B, ..., AB

Complement of Event Black,

Black : 2R, 2R, ..., AR, AR

Sample

Space:

2R, 2R,

2B, ..., AB

Experiment: Draw 1 Card. Note Number,

Color & Suit.


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Experiment: Draw 1 Card. Note Suit. A: , , B: ,

C:,

Example

1. What is the relationship between events Band C?

2. AC

:3. BC:4. BC:

Mutually exclusive; B = CC

,,,


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Experiment: Roll 1 Die. A: Even number B: { 1, 3} C: {2, 3, 6}

Example

1. What is the relationship between events A and B?

2. BC

:3. BC:4. BC:

Mutually exclusive;

{2, 4, 5, 6}{ 3 }

{1, 2, 3, 6}


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Select a student from the classroom and recordhis/herhair colorandgender. A: student has brown hair B: student is female

C:student is male

Example

1. What is the relationship between events Band C?

2. AC

:3. BC:4. BC:

Mutually exclusive; B = CC

Student does not have brown hairStudent is both male and female =

Student is either male and female = all students = S


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Which of the following pairs of sets are mutually

exclusive events when a single card is chosen atrandom from a standard deck of 52 playing cards?

1. Choosing a 7 or choosing a club.

2. Choosing a 7 or choosing a jack.3. Choosing a 7 or choosing a heart.

4.None of the above.

Example


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Probabilities


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Thinking Challenge

Whats the probabilityof getting a headonthe toss of a single fair

coin? Use a scale from0(noway) to 1(surething).

So toss a coin twice.

Do it! Did you get onehead & one tail?Whats it all mean?


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http://bcs.whfreeman.com/ips4e/cat_010/applets/Pro

bability.html

Example
http://bcs.whfreeman.com/ips4e/cat_010/applets/Probability.htmlhttp://bcs.whfreeman.com/ips4e/cat_010/applets/Probability.htmlhttp://bcs.whfreeman.com/ips4e/cat_010/applets/Probability.htmlhttp://bcs.whfreeman.com/ips4e/cat_010/applets/Probability.html


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Many Repetitions!

Number of Tosses

Total Heads /Number of Tosses

0.00

0.25

0.50

0.75

1.00

0 25 50 75 100 125


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In Chapters 1,2, we used graphs and numerical measuresto describe data sets which were usuallysamples.

We measured how often using

What is Probability?

Relative frequency =

SampleAnd How often= Relative frequency

Population

Probability

As ngets larger,


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= lim

The probability of an event A measures how often

we think A will occur. We write P(A).

The Probability of an Event


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What is Probability?

1. Numerical Measureof Likelihood thatEvent Will Occur P(Event)

P(A)

Prob(A)

2. Lies Between 0 & 13. Sum of outcome

probabilities is 1

1

.5

0

Sure

Impossible


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P(A) must be between 0 and 1.

If event A can never occur, P(A) = 0. If event A alwaysoccurs when the experiment is performed, P(A) =1.

The sum of the probabilities for all simple events in S equals 1.

The probability of an event A is found by adding theprobabilities of all the simple events contained in A.

The probability of an event A is found by adding theprobabilities of all the simple events contained in A.

The Probability of an Event


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A teacher chooses a student at random from a class

of 30 girls. What is the probability that the studentchosen is a girl?

A teacher chooses a student at random from a classof 30 girls. What is the probability that the student

chosen is a boy?

Example


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If the simple events in an experiment are equally

likely, you can calculate

=

=

Counting Rules


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Toss a fair coin twice. What is the probability ofobserving at least one head?

Example

H

1st Coin 2nd Coin Ei P(Ei)

H

T

T

H

T

HH

HT

TH

TT

1/4

1/4

1/4

1/4

P(at least 1 head)

= P(E1) + P(E2) + P(E3)

= 1/4 + 1/4 + 1/4 = 3/4


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A bowl contains three M&Ms, one red, one blue

and one green. A child selects two M&Ms atrandom. What is the probability that at least one isred?

Example

1st M&M 2nd M&M Ei P(Ei)RB

RG

BR

BG

1/6

1/6

1/6

1/6

1/6

1/6

P(at least 1 red)

= P(RB) + P(BR)+ P(RG)

+ P(GR)

= 4/6 = 2/3

m

m

m

m

m

m

m

m

mGB

GR


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Example

A card is drawn from a well-shuffled deck of 52 playingcards. What is the probability that it is a queen ora heart?

Let Q= Queen and H= Heart

=

, =

,

=

= + =

+

=


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Consider a student that attends Carleton. Suppose

the probability that the student is from Ottawa is .25. What is the probability that the student is not from

Ottawa?

Example


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A single card is chosen at random from a standard

deck of 52 playing cards. What is the probability ofchoosing a card that is not a king?

If a single 6-sided die is rolled, what is the probabilityof rolling a number that is not 8?

Example


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A sample of four students was asked to toss a fair coin sixtimes and record each result as either heads (H) or tails(T).

Student Sequence of 6 H's and T's1 H T T T H H

2 T T T T T T

3 H T H T H T4 T T T H H H

Do the results from each the four students all have thesame probability of happening?

Thinking Challenge


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Self-Correcting Exercises

Section 2.1: 1, 3, 5, 7, 9.

Section 2.2: 11, 13,15, 17, 19.

Practical Questions

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Probability Basics (1)