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Copyright 2004 Pearson Education, Inc.

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Chapter 3Probability

3-1 Overview

3-2 Fundamentals

3-3 Addition Rule

3-4 Multiplication Rule: Basics

3-5 Multiplication Rule: Complements and

Conditional Probability3-6 Probabilities Through Simulations

3-7 Counting

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Created by Tom Wegleitner, Centreville, Virginia

Section 3-1 & 3-2Overview & Fundamentals

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Overview

Rare Event Rule for Inferential Statistics:

If, under a given assumption (such as a lottery beingfair), the probability of a particular observed event(such as five consecutive lottery wins) is extremelysmall, we conclude that the assumption is probablynot correct.

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Definitions

Event

Any collection of results or outcomes of aprocedure.

Simple Event

An outcome or an event that cannot be furtherbroken down into simpler components.

Sample SpaceConsists of all possible simple events. That is,the sample space consists of all outcomes thatcannot be broken down any further.

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Notation forProbabilities

P- denotes a probability.

A, B, and C- denote specific events.

P(A) - denotes the probability ofevent A occurring.

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Basic Rules forComputing Probability

Rule 1: Relative Frequency Approximationof Probability

Conduct (or observe) a procedure a large

number of times, and count the number oftimes eventA actually occurs. Based onthese actual results, P(A) is estimatedasfollows:

P(A) = number of times A occurrednumber of times trial was repeated

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Rule 2: Classical Approach to Probability(Requires Equally Likely Outcomes)

Assume that a given procedure has n different

simple events and that each of those simpleevents has an equalchance of occurring. IfeventA can occur in s of these n ways, then

P(A) = number of waysA can occurnumber of different

simple events

s

n=

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Rule 3: Subjective Probabilities

P(A), the probability of eventA, is found bysimply guessing or estimating its value

based on knowledge of the relevant

circumstances.

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Law of

LargeN

umbersAs a procedure is repeated again and

again, the relative frequency probability(from Rule 1) of an event tends to

approach the actual probability.

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Example

Roulette You plan to bet on number 13 on the next spinof a roulette wheel. What is the probability that you willlose?

Solution A roulette wheel has 38 different slots, only

one of which is the number 13. A roulette wheel isdesigned so that the 38 slots are equally likely. Amongthese 38 slots, there are 37 that result in a loss.Because the sample space includes equally likelyoutcomes, we use the classical approach (Rule 2) to get

P(loss) =37

38

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Probability Limits

The probability of an event that is certain tooccur is 1.

The probability of an impossible event is 0.

0 e P(A) e 1 for any eventA.

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Possible Values forProbabilities

Figure 3-2

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Definition

The complement of eventA, denoted by

A, consists of all outcomes in which theeventA does notoccur.

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Example

Birth Genders In reality, more boys are born thangirls. In one typical group, there are 205 newbornbabies, 105 of whom are boys. If one baby israndomly selected from the group, what is the

probability that the baby is nota boy?

Solution Because 105 of the 205 babies are boys, it follows that100 of them are girls, so

P(not selecting a boy) = P(boy) = P(girl) 100 0.488205

! !

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Rounding OffProbabilities

When expressing the value of a probability,either give the exactfraction or decimal orround off final decimal results to threesignificant digits. (Suggestion: When theprobability is not a simple fraction such as 2/3or 5/9, express it as a decimal so that the

number can be better understood.)

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Definitions

The actual odds against eventA occurring are the ratioP(A)/P(A), usually expressed in the form ofa:b (or a

to b), where a and b are integers having no common

factors.

The actual odds in favoreventA occurring are thereciprocal of the actual odds against the event. If the

odds againstA are a:b, then the odds in favor ofA are

b:a.

The payoff odds against eventA represent the ratio of

the net profit (if you win) to the amount bet.

payoff odds against event A = (net profit) : (amount bet)

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Recap

In this section we have discussed:

Rare event rule for inferential statistics.

Probability rules.

Law of large numbers.

Complementary events.

Rounding off probabilities.

Odds.

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Section 3-3Addition Rule

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Compound Event

Any event combining 2 or more simple events

Definition

Notation

P(A orB) = P(eventA occurs or

event B occurs or they bothoccur)

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When finding the probability that eventA

occurs or event B occurs, find the total

number of waysA can occur and thenumber of ways B can occur, but findthe

totalinsuch a waythatnooutcomeis

countedmorethanonce.

General Rule for aCompound Event

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Compound Event

Intuitive Addition Rule

To find P(A orB), find the sum of the number of

ways eventA can occur and the number of ways event Bcan occur, addinginsuch a waythateveryoutcomeis

countedonlyonce. P(A orB) is equal to that sum,

divided by the total number of outcomes. In the sample

space.

Formal Addition Rule

P(A orB) = P(A) + P(B) P(A and B)

where P(A and B) denotes the probability that A

and B both occur at the same time as an outcome in

a trial or procedure.

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Definition

Figures 3-4 and 3-5

EventsA and B are disjoint (ormutuallyexclusive) if they cannot both occurtogether.

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Applying the Addition Rule

Figure 3-6

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Find the probability of randomly selecting a man or a boy.

Titanic PassengersMen Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Adapted from Exercises 9 thru 12

Example

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Find the probability of randomly selecting a man or a boy.

P(man or boy) = 1692 + 64 = 1756 = 0.7902223 2223 2223

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

* Disjoint *

Example


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Find the probability of randomly selecting a man or

someone who survived.

Men Women Boys Girls TotalsSurvived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

Example


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Find the probability of randomly selecting a man or

someone who survived.

P(man or survivor) = 1692 + 706 332 = 20662223 2223 2223 2223

Men Women Boys Girls TotalsSurvived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

* NOT Disjoint *

= 0.929


Example

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ComplementaryEvents

P(A) and P(A)

are

mutually exclusive

All simple events are either inA orA.

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P(A) + P(A) = 1

= 1 P(A)

P(A) = 1 P(A)

P(A)

Rules ofComplementary Events

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Venn Diagram for theComplement of Event A

Figure 3-7

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Recap


Compound events.

Formal addition rule.

Intuitive addition rule.

Disjoint Events.

Complementary events.

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Section 3-4Multiplication Rule:

Basics

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Notation

P(A and B) =

P(eventA occurs in a first trial andevent B occurs in a second trial)

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Tree DiagramsA tree diagram is a picture of the possible outcomes of a

procedure, shown as line segments emanating from one

starting point. These diagrams are helpful in counting the

number of possible outcomes if the number of possibilities is

not too large.

Figure 3-8 summarizes

the possible outcomes

for a true/false followed

by a multiple choice question.

Note that there are 10 possible combinations.

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ExampleGenetics Experiment Mendels famous hybridization

experiments involved peas, like those shown in Figure 3-3(below). If two of the peas shown in the figure are

randomly selected withoutreplacement, find the

probability that the first selection has a green pod and the

second has a yellow pod.

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Example - Solution

First selection: P(green pod) = 8/14 (14 peas, 8 of which have

green pods)

Second selection: P(yellow pod) = 6/13 (13 peas remaining, 6

of which have yellow pods)

With P(first pea with green pod) = 8/14 and P(second pea with

yellow pod) = 6/13, we have

P( First pea with green pod and second pea with yellow pod) =

8 6. 64

14 13y }

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ExampleImportant Principle

The preceding example illustrates the

important principle that the probabilityforthe

secondeventB shouldtakeintoaccountthe

factthatthe firsteventA hasalreadyoccurred.

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Notation forConditional Probability

P(B A) represents the probability of event Boccurring after it is assumed that eventA hasalready occurred (read B A as B givenA.)

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Definitions

Independent Events

Two eventsA and B are independent if theoccurrence of one does not affect the

probability of the occurrence of the other.(Several events are similarly independent if theoccurrence of any does not affect theoccurrence of the others.) IfA and B are not

independent, they are said to be dependent.

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FormalMultiplication Rule

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

Note that if A and B are independent

events, P(B A) is really the same as

P(B)

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Applying theMultiplication Rule

Figure 3-9

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Small Samples fromLarge Populations

If a sample size is no more than 5% of

the size of the population, treat the

selections as being independent(evenif the selections are made without

replacement, so they are technically

dependent).

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Summary of Fundamentals

In the addition rule, the word or on P(A orB)

suggests addition. Add P(A) and P(B), being

careful to add in such a way that every outcome is

counted only once.

In the multiplication rule, the word and in P(A and B)

suggests multiplication. Multiply P(A) and P(B),

but be sure that the probability of event B takesinto account the previous occurrence of eventA.

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Recap


Notation forP(A and B).

Notation for conditional probability.

Independent events.

Formal and intuitive multiplication rules.

Tree diagrams.

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Section 3-5Multiplication Rule:

Complements and

Conditional Probability

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Complements: The Probabilityof At Least One

The complement of getting at least oneitem of a particular type is that you get

no items of that type.

At least one is equivalent to one ormore.

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Example

Gender of Children Find the probability of a couplehaving at least 1 girl among 3 children. Assume thatboys and girls are equally likely and that the gender ofa child is independent of the gender of any brothers or

sisters.

Solution

Step 1: Use a symbol to represent the event desired.

In this case, letA = at least 1 of the 3children is a girl.

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Solution (cont)

Step 2: Identify the event that is the complement ofA.

A = not getting at least 1 girl among 3children

= all 3 children are boys

= boy and boy and boy

Example

Step 3: Find the probability of the complement.

P(A) = P(boy and boy and boy)

1 1 1 12 2 2 8

! y y !

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Example

Solution (cont)

Step 4: Find P(A) by evaluating 1 P(A).

1 7( ) 1 ( ) 18 8

! ! !P A P A

Interpretation There is a 7/8 probability that if acouple has 3 children, at least 1 of them is a girl.

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Key Principle

To find the probability ofatleastone ofsomething, calculate the probability ofnone, then subtract that result from 1.

That is,

P(at least one) = 1 P(none)

f

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Definition

A conditional probability of an event is a probabilityobtained with the additional information that some otherevent has already occurred. P(B A) denotes theconditional probability of event B occurring, given thatA

has already occurred, and it can be found by dividing theprobability of eventsA and B both occurring by theprobability of eventA:

P(B A) =

P(A and B)

P(A)

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Testing for Independence

In Section 3-4 we stated that eventsA and B areindependent if the occurrence of one does notaffect the probability of occurrence of the other.This suggests the following test for independence:

Two eventsA and B areindependentif

Two eventsA and B aredependentif

P(B A) = P(B)

or

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

P(B A) = P(B)

or

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

R

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Recap


Concept of at least one.

Conditional probability.

Intuitive approach to conditional probability.

Testing for independence.

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Section 3-6Probabilities Through

Simulations

D fi iti

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Definition

A simulation of a procedure is a

process that behaves the same way asthe procedure, so that similar results

are produced.

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Si l ti E l

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Simulation Examples

Solution 1:

Flipping a fair coin where heads = female and

tails = male

H H T H T T H H H H

female female male female male male male female female female

Si l ti E l

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Simulation Examples

Solution 1:

Flipping a fair coin where heads = female andtails = male

H H T H T T H H H H

female female male female male male male female female female

Solution2:

Generating 0s and 1s with a computer or

calculator where 0 = male1 = female

0 0 1 0 1 1 1 0 0 0

male male female male female female female male male male

R d N b

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Random Numbers

In many experiments, random numbers are used inthe simulation naturally occurring events. Below are

some ways to generate random numbers.

A random table of digits

Minitab

Excel

TI-83 Plus calculator

STATDISK

R

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Recap


The definition of a simulation.

Ways to generate random numbers.

How to create a simulation.

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Section 3-7Counting

F d t l C ti R l

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Fundamental Counting Rule

For a sequence of two events in which

the first event can occurm ways and

the second event can occurn ways,

the events together can occur a total of

m n ways.

N t ti

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Notation

The factorial symbol ! Denotes the product of

decreasing positive whole numbers. For example,

! .! y y y !

By special definition, 0! = .

F t i l R l

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A collection ofn different items can be

arranged in ordern! different ways.

(This factorial rule reflects the fact thatthe first item may be selected in n

different ways, the second item may be

selected in n 1 ways, and so on.)

Factorial Rule

P t ti R l

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Permutations Rule(when items are all different)

(n

-r

)!

n rP =n!

The number ofpermutations (or sequences)

ofr items selected from n available items

(without replacement is

Permutation Rule:

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Permutation Rule:

Conditions

We must have a total ofn different items

available. (This rule does not apply if

some items are identical to others.)

We must select rof the n items (without

replacement.)

We must consider rearrangements of thesame items to be different sequences.

P t ti R l

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Permutations Rule( when some items are identical to others )

n1! . n2! .. . . . . . . nk!

n!

If there are n items with n1 alike, n2 alike, .

. . nk alike, the number of permutations of

all n items is

C bi ti R l

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(n - r )! r!n!

nC

r=

Combinations Rule

The number of combinations ofr items

selected from n different items is

Combinations Rule:

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Combinations Rule:

Conditions

We must have a total ofn different items

available.

We must select rof the n items (without

replacement.)

We must consider rearrangements of thesame items to be the same. (The

combinationABCis the same as

CBA.)

Permutations versus

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When different orderings of the same items

are to be counted separately, we have a

permutation problem, but when differentorderings are not to be counted separately,

we have a combination problem.

Permutations versus

Combinations

Recap

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Slide 74Recap


The fundamental counting rule.

The permutations rule (when items are alldifferent.)

The permutations rule (when some items areidentical to others.)

The combinations rule.

The factorial rule.

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