Download Probabilities for

Probability
and
Counting Rules
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Introduction
4-1 Sample Spaces and Probability
4-2 Addition Rules for Probability
4-3 Multiplication Rules & Conditional Probability
4-4 Counting Rules
4-5 Probability and Counting Rules
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Sample Spaces and Probability
Probability can be defined as the chance of an event
occurring
A probability experiment is a chance process that
leads to well-defined results called outcomes.
 An outcome is the result of a single trial of a
probability experiment.
 A sample space is the set of all possible outcomes of
a probability experiment. The symbol ( S ) is used for
the sample space .
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Some Sample Spaces
Experiment
Sample Space
Toss a coin
S={Head , Tail}
Roll a die
S={1, 2, 3, 4, 5, 6}
Answer a true/false question
Toss two coins
S={True, False}
S={HH, HT, TH, TT}
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Example 4-1: Rolling Dice
Find the sample space for rolling two dice.
N(S) = 62
Example 4-3: Gender of Children
Find the sample space for the gender of the children
if a family has three children. Use B for boy and G
for girl.
Solution :
2n=23 = 8
S={BBB , BBG , BGB , BGG , GBB , GBG , GGB
,GGG}
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A tree diagram is a device consisting of line segments
emanating from a starting point and also from the outcome
point .It is used to determine all possible outcomes of a
probability experiment.
Example 4-4: Gender of Children
Use a tree diagram to find the sample space for the
gender of three children in a family,
as in example 4-3.
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Solution :
B BBB
B
B
G
G BBG
B BGB
G
BGG
B GBB
B
G
G
G GBG
B GGB
G
GGG
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 An event consists of outcomes of a probability
Experiment .)‫(عناصرها مجموعة جزئية من فراغ العينة ونرمز لها بحروف كبيرة‬
event
Simple event
is an event with
one outcome.
For example :
Compound event
is an event with
containing more
than one outcome
S={1,2,3,4,5,6}
A = { 6 } Simple event
B = Odd no. = { 1 , 3 , 5 } Compound event
E = Even no. = { 2 , 4 , 6 } Compound event
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 Types of probability:

Classical Probability

Empirical Probability or Relative Frequency.

Subjective Probability
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Classical probability
 Classical probability uses sample spaces to determine the
numerical probability that an event will happen and assumes
that all outcomes in the sample space are equally likely to
occur.
 Equally Likely Events are events that have the same probability
of occurring
nE
# of desired outcomes
PE 

n  S  Total # of possible outcomes
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Example 4-6: Gender of Children
If a family has three children, find the probability that two of
the three children are girls.
Solution :
Step 1 : Sample Space:
S ={BBB ,BBG, BGB, BGG, GBB ,GBG ,GGB ,GGG}
Step 2 : k={BGG, GBG, GGB}
P(K)=
= 3/8
The probability of having two of three children being girls
is 3/8.
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Probability Rules
There are four basic probability rules:

First Rule: For any event E 0 ≤ P(E) ≤ 1
)‫(االحتمال عدد كسري اوعشري مستحيل يكون سالب او اكبر من واحد‬

Second Rule: If an event E cannot occur , then P(E)= 0.
 Third
Rule: If an event E is certain to occur, then P(S) =1.
Fourth
Rule: The sum of the probabilities of all the outcomes
in the sample space is 1 ∑ p = 1
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Example 4-8: Rolling a Die
When a single die is rolled , find the probability of getting a 9 .
Solution :
S= {1 , 2 , 3 , 4 , 5 , 6 } , A={ø}
P(9) =
= 0/6 = 0
Second Rule
Example 4-9:
When a single die is rolled ,what is the probability of getting a
number less than 7 ?.
Solution :
S={1 , 2 , 3 , 4 , 5 , 6} , A ={1 , 2 , 3 , 4 , 5 , 6}
P(A)=
= 6/6 = 1
Fourth Rule
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The complement of an event E , denoted by
, is the set of
outcomes in the sample space that are not included in the
outcomes of event E.
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Example 4-10:
Find the complements of each event.
Event ( E )
Rolling a die and getting a 4
Selecting a month and getting a
month that begins with a J
Selecting a day of the week and
getting a weekday
Complement of Event ( E )
Getting a 1, 2, 3, 5, or 6
Getting February, March, April, May,
August, September, October, November,
or December
Getting Friday or Saturday
Selecting a day of the week that Getting Sunday,Monday,Wednesday,Friday
beings with the letter T
and Saturday
Rolling two dice and getting
A sum of 2, 4, 6, 8,10 or 12
a sum that is odd no.
P (S)
P (E)
P (E)
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Rule for Complementary Events
Example :
If the probability that a person lives in an industrialized
country of the world is , Find the probability that a person
does not live in an industrialized country.
P Not living in industrialized country 
= 1  P  living in industrialized country 
1 4
 1 
5 5
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Example 4-11:
In a study, it was found that 23% of the people surveyed
said that vanilla was their favorite flavor of ice cream. If
a person is selected at random. Find the probability that
the person’s favorite of ice cream is not vanilla.
Solution:
P( not vanilla) = 1 – P(vanilla)
=1 – 0.23 = 0.77 = 77%
Empirical Probability- Relative Frequency
Empirical probability relies on actual experience to
determine the likelihood of outcomes.
f frequency of desired class
PE  
n
Sum of all frequencies
Remark:
 or indicate the Union ( + ).[or =  = +]
 and indicate intersection ( × ). [and =  = × ]
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Example 4-13:
In a sample of 50 people, 21 had type O blood, 22 had type A
blood, 5 had type B blood, and 2 had type AB blood. Set up a
frequency distribution and find the following probabilities.
a. A person has type O blood.
Type
Frequency
A
22
B
5
AB
2
O
21
Total
f
P O 
n
21

50
50
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b. A person has type A or type B blood.
Type
A
B
AB
O
Total
Frequency
22
5
2
22 5

50 50
27

50
P  A or B 
21
50
c. A person has neither type A nor type O blood.
Type
A
B
AB
O
Total
Frequency
22
5
2
21
50
P  neither A nor O 
5
2

50 50
7

50

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d. A person does not have type AB blood.
Type
A
B
Frequency
22
5
AB
O
Total
2
21
50
P  not AB 
 1  P  AB 
2 48 24
 1


50 50 25
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Subjective probability
Subjective probability uses a probability value based on an
educated guess or estimate, employing opinions and inexact
information.
Examples: weather forecasting, predicting outcomes of
sporting events
A sportswriter may say that there is a 70% probability that the Pirates will
win the pennant next year.
A physician might say that, on the basis of her diagnosis, there is a 30%
chance the patient will need an operation.
A seismologist might say there is an 80% probability that an earthquake
will occur in a certain area.
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‫‪Subjective‬‬
‫مبني على التخمين والتوقع بناء على خبرة الشخص‬
‫مثل‪ :‬توقع االرصادي بكمية االمطار او خبير الزالزل والبراكين بالزالزل‬
‫‪‬‬
‫والبراكين‪,‬المحلل الرياضي بنتائج مبارايات او بفوز حصان معين في‬
‫السباق او تشخيص اخصائي لفعالية دواء‪...‬الخ‬
‫‪‬‬
‫يشير للمستقبل ‪Next‬‬
Questions ???
1- a jellybean is chosen at random from a jar containing 5 black , 8
red and 7 yellow jellybeans . Find the probability that it is :
a) red
b) yellow
c) not black
2- If A ,B and C three events , the probability of the events A and B is
P(A)= 0.40 , P(B)= 0.37 .Find P(C) ?
3- A probability experiment is conducted. Which of these cannot be
considered a probability of an event?
a) 0.75
b) 0
c) 1
d) -0.25
4- Classify each statement as an example of Classical probability
, Empirical probability , Subjective probability :
a) The probability that a person will watch the 6 o’clock
evening news is 0.15.
Empirical probability
b) The probability of wining at a chuck – a – luck game is
Classical probability
c) The probability that a bus will be in an accident on a specific
run is about 6% .
Empirical probability
Page 198
Exercises 4-1
Pag 198
(10,11,12,15,21,22,25)
Addition Rules for Probability
 Two events are Mutually Exclusive Events if
they cannot occur at the same time (i.e., they have
no outcomes in common)
P(A or B)=P(A) + P(B) Mutually Exclusive
 This means that P)A∩B(= 0
i.e. the two event cannot occur
at the same time .
P (S)
B
A
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 Two events are Not Mutually Exclusive Events,
then the probability of event A or B occurs denoted
by P(AUB), is given by
P(AUB)= P(A) + P(B) – P(A∩B) Not mutually exclusive
P)A∩B(
 Where P(A∩B) is
the probability both A
and B occur.
P (S)
B
A
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Example 4-15: Rolling a Die
Determine which events are mutually exclusive and which
are not, when a single die is rolled.
a. Getting an odd number and getting an even number
Getting an odd number: 1, 3, or 5
Getting an even number: 2, 4, or 6
Mutually Exclusive
b. Getting a 3 and getting an odd number
Getting a 3: 3
Getting an odd number: 1, 3, or 5
Not Mutually Exclusive
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c. Getting an odd number and getting a number less than 4
Getting an odd number: 1, 3, or 5
Getting a number less than 4: 1, 2, or 3
Not Mutually Exclusive
d. Getting a number greater than 4 and getting a number
less than 4
Getting a number greater than 4: 5 or 6
Getting a number less than 4: 1, 2, or 3
Mutually Exclusive
Example 4-15: see page 202
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Example :
Selecting a Doughnut
A box contains 3 glazed doughnuts , 4 jelly doughnuts , and 5
chocolate doughnuts. If a person selects a doughnut at random
,find the probability that it is either a glazed doughnut or a
chocolate doughnut.
Solution :
The events
P(glazed) + P(chocolate) =
are mutually
exclusive
Example :
Solution :
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Example 4-19: page 204
8% like cookies and cream, and 6% like mint chocolate chip
Find P(cookies and cream or mint chocolate chip) = 0.08 + 0.06
= 0.14 = 14%
Example 4-21:
In a hospital unit there are 8 nurses and 5 physicians ;7 nurses
and 3 physicians are females. If a staff person is selected ,find
the probability that the subject is a nurse or a male.
Staff
Nurses
Physicians
Total
Females
7
3
10
Males
1
2
3
Total
8
5
13
Solution :
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Which one of these events is not mutually exclusive?
A) Select a student in your university: The student is married, and
the student is a business major.
B) Select a ball from bag: It is a football, and it is a basket ball.
C) Roll a die: Get an even number, and get an odd number.
D) Select any course: It is an Arabic course, and it is a Statistics
course.
e) Select a student in your university: The student is female, and the
student is a junior.
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Exercises 4-2
page 206
3,4,5,9,11,13,24
The Multiplication Rules and
Conditional Probability
Multiplication Rules
 Two events A and B are independent events if the
fact that A occurs does not affect the probability of B
occurring.
P(A and B)=P(A) . P(B) Independent Events
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Example 4-25: Selecting a Colored Ball
An urn contains 3 red balls , 2blue balls and 5 white balls .A ball is
selected and its color noted .Then it is replaced .A second ball is selected
and its color noted . Find the probability of each of these.
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Example 4-27: Male Color Blindness
Approximately 9% of men have a type of color blindness
that prevents them from distinguishing between red and
green . If 3 men are selected at random , find the probability
that all of them will have this type of red-green color
blindness.
Solution :
Let C denote red – green color blindness. Then
P(C and C and C) = P(C) . P(C) . P(C)
= (0.09)(0.09)(0.09)
= 0.000729
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 When the outcome or occurrence of the first event
affects the outcome or occurrence of the second event
in such a way that the probability is changed ,the
events are said to be dependent events.
P(A and B)=P(A) . P(B|A) dependent Events
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Example 4-28: University Crime
At a university in western Pennsylvania, there were 5
burglaries reported in 2003, 16 in 2004, and 32 in 2005. If a
researcher wishes to select at random two burglaries to
further investigate, find the probability that both will have
occurred in 2004.
Solution :
Dependent Events
P  C1 and C2   P  C1   P  C2 C1 
16 15
60



53 52
689
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Example 4-29: Homeowner’s and Automobile Insurance
World Wide Insurance Company found that 53% of the residents
of a city had homeowner’s insurance )H( with the company .Of
these clients ,27% also had automobile insurance (A) with the
company .If a resident is selected at random ,find the probability
that the resident has both homeowner’s and automobile
insurance with World Wide Insurance Company .
Solution :
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If P(A) = 0.3, P(B) = 0.4 , and A,B are mutually exclusive
events, find P(A and B).
a) 0
b) 1
c) 0.12
d) 0.7
If P(A) = 0.2, P(B) = 0.4 , and A,B are independent events,
find P(A and B).
a) 0
b) 1
c) 0.12
d) 0.08
Example 4-31: Selecting Colored Balls
Box 1 contains 2 red balls and 1 blue ball . Box 2 contains 3
blue balls and 1 red ball . A coin is tossed . If it falls heads
up ,box1 is selected and a ball is drawn . If it falls tails up
,box 2 is selected and a ball is drawn. Find the probability of
selecting a red ball.
Box 1
Box 2
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Solution :
Red
Box 1
Blue
Coin
Red
Box 2
Blue
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A red ball can be selected from either box1 or box2
P(red)=
•Box A contains 4 red balls and 2 white balls. Box B contains 2 red
balls, 2 white balls. A die is rolled first and if the outcome is an even
number a ball is chosen at random from Box A, and if the outcome is
an odd number a ball is randomly chosen from Box B.
Find the probability that a red ball is chosen?
2
9
7
12
6
24
2
12
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*box1 contains 20% defective transistors,box2 contains 30%
defective transistors, and box3 contains 50% defective
transistors. A die is rolled. If the number that appears is
greater than 3, a transistor is selected from 1. If the
number is less than 3, a transistor is selected from 2. If the
number is 3, a transistor is selected from box3 . Find the
probability of selecting a defective transistor.
a) 0.028
b) 1
c) 0.283
d) 0.03
* A die is rolled. What is the probability that the number
rolled is greater than 2 and even number?
1/3
Conditional Probability
 Conditional probability is the probability that
the second event B occurs given that the first event
A has occurred.
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Example 4-32: Selecting Colored Chips
A box contains black chips and white chips. A person selects two chips
without replacement . If the probability of selecting a black chip and a
white chip is
, and the probability of selecting a black chip on the first
draw is
, find the probability of selecting the white chip on the
second draw ,given that the first chip selected was a black chip.
Solution :
Let
B=selecting a black chip W=selecting a white chip
Hence , the probability of selecting a while chip on the second
draw given that the first chip selected was black is
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Example 4-34: Survey on Women In the Military
A recent survey asked 100 people if they thought women in the armed
forces should be permitted to participate in combat. The results of the
survey are shown.
a. Find the probability that the respondent answered yes (Y),
given that the respondent was a female (F).
8
P  F and Y 
8
4
100
P YF 
 50 

P  F
50 25
100


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b. Find the probability that the respondent was a male
(M), given that the respondent answered no (N).
18
P  N and M 
100
P MN 

60
P  N
100


18
3


60 10
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Male
Female
Total
Doctor
4
12
16
Physiotherapy
6
14
20
Nurse
9
11
20
Total
19
37
56
•Find the probability that the employee is a doctor or a female.
•Find the probability that the employee is a physiotherapy.
•Find the probability that the employee is a nurse and a male.
•Find the probability that the employee is a doctor given that he
a male .
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Probabilities for (At Least)
Example 4-36: Tossing Coins
A coin is tossed 5 times . Find the probability of getting at least 1 tail.
P( at least 1 tail ) = 1 – P( all heads) = 1 – P(HHHHH)
=1= 1Note: This PowerPoint is only a summary and your main source should be the book.
For Example: It has been found that 8% of all automobiles
on the road have defective brakes. If 8 automobiles are
stopped and checked by the police ,find the probability that
at least one will have defective brakes.
Solution :
P(at least one will have defective brakes) = 1 – p( all have not defective brakes)
= 1- (1- 0.08)8
= 1- (0.92)8
= 0.487
Example 4-37: see page 221
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It is reported that 27% of working women use computers at
work. Choose 5 working women at random .
1)
2)
3)
Find the probability that at least 1 use a computer at
work.
Find the probability that at least 1 doesn’t use a
computer at work.
Find the probability that all 5 use a computer in their
jobs.
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Exercises 4-3
page 222
1,2,3,6,8,16,20,27,30,35,37,41,
45,51
Counting Rules
Fundamental Counting Rule
 The Fundamental Counting Rule is also called
the multiplication of choices.
 In a sequence of n events in which the first one
has k1 possibilities and the second event has k2 and
the third has k3, and so forth, the total number of
possibilities of the sequence will be
k1 · k2 · k3 · · · kn
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Example 4-38:
A coin is tossed and die is rolled . Find the number of
outcomes for the sequence of events.
Example 4-39: Paint Colors
A paint manufacturer wishes to manufacture several different paints. The
categories include
Color: red, blue, white, black, green, brown, yellow
Type: latex, oil
Texture: flat, semi gloss, high gloss
Use: outdoor, indoor
How many different kinds of paint can be made if you can select one
color, one type, one texture, and one use?
Solution :

 
 
# of
# of
# of
# of
colors types textures uses
7

2

3

2
84 different kinds of paint
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 Factorial is the product of all the positive
numbers from 1 to a number.
n !  n  n  1 n  2   3  2 1
0!  1
 Permutation is an arrangement of objects in a
specific order. Order matters.
n!  n  n  1 n  2   n  r  1
n Pr 
 n  r !
r items
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Example 4-41:
The first year the state of Pennsylvania issued railroad
memorial license plates, the plates had a picture of a steam
engine followed by four digits. Assuming that repetitions are
allowed, how many railroad memorial plates could be issued?
If repetitions are not allowed?????
Solution :
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Example 4-42: Business Locations
Suppose a business owner has a choice of 5 locations in which
to establish her business. She decides to rank each location
according to certain criteria, such as price of the store and
parking facilities. How many different ways can she rank the 5
locations?
Solution :





first second third fourth fifth
choice choice choice choice choice

120 different ways to rank the locations
Using factorials, 5! = 120.
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Example 4-44: see page 231
Example : News Stories
A television news director wishes to use 3 news stories on an
evening show. One story will be the lead story, one will be
the second story, and the last will be a closing story. If the
director has a total of 8 stories to choose from, how many
possible ways can the program be set up?
Solution :
Since there is a lead, second, and closing story, we know that
order matters. We will use permutations.
8!
 336
8 P3 
5!
or
P  8  7  6  336
8 3
3
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 Combination is a grouping of objects. Order
does not matter.
n!
n Cr 
 n  r  !r !
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Example 4-47: Combinations
How many combinations of 4 objects are there . Taken 2 at a
time?
Solution :
This is a combination problem , the answer is
4c2
Example 4-48:see page 233
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Example 4-49: Committee Selection
In a club there are 7 women and 5 men. A committee of 3
women and 2 men is to be chosen. How many different
possibilities are there?
Solution :
There are not separate roles listed for each committee member,
so order does not matter. We will use combinations.
7!
5!
Women: 7C3 
 35, Men: 5C2 
 10
4!3!
3!2!
There are 35·10=350 different possibilities.
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Summary of Counting Rules
Exercises 4-4
page 235
1, 3,12,17,19, 21, 23, 43,59,63
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Probability and Counting Rules
Example 4-51: Defective Transistors
A box contains 24 transistors ,4 of which are defective. If 4
are sold at random ,find the following probabilities.
Solution :
a. Exactly 2 are defective.
Def
4
2
2
Non
def
20
24
4
Note: This PowerPoint is only a summary and your main source should be the book.
b. None is defective
Def
4
0
4
Non
def
20
24
4
Def
4
c. All are defective.
4
0
Non
def
20
24
4
d. At least 1 is defective.
Note: This PowerPoint is only a summary and your main source should be the book.
Example52 : Committee Selection
A store has 6 TV Graphic magazines and 8 News time
magazines on the counter. If two customers purchased a
magazine, find the probability that one of each magazine
was purchased.
Solution :
TV . G
6
1
1
6
News
14
time
8
2
C1 8 C1 6  8
48


91
91
14 C 2
Note: This PowerPoint is only a summary and your main source should be the book.
Example : Combination Locks
A combination lock consists of the 26 letters of the alphabet.
If a 3-letter combination is needed, find the probability that
the combination will consist of the letters ABC in that order.
The same letter can be used more than once. (Note: A
combination lock is really a permutation lock.)
Solution :
1/26
1/26
1/26
Note: This PowerPoint is only a summary and your main source should be the book.
Example 4-54: Tennis Tournament
There are 8 married couples in a tennis club. If 1 man and 1
woman are selected at random to plan the summer
tournament ,find the probability that they are married to each
other.
Solution :
There are 8 ways to select the man and 8 ways to select the
woman ,there are 8.8 or 64 ,ways to select 1 man and 1
woman .
Note: This PowerPoint is only a summary and your main source should be the book.
•A store contains 10 toys, 6 of which are defective. If a child bought 3
toys at random, what is the probability of getting 2 defective toys?
• A person owns a collection of 25 movies, five of which are
English. If four movies are selected at random, find the probability
that three of them are English.
• Given nine flowers, four of which are white and five of
them are red, if two flowers are selected at random, without
replacement, what is the probability that both flowers are
red?
•The probability of randomly selecting 3 science books and 4 math
books from 8 science books and 9 math books is
• If 6 different-size washers are arranged in arrow, then the
probability that they will be arranged in order of size
Exercises 4-5
page 245
3,6,7,11(a-b-c),12
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