Transcript Event

Chapter # 4
Probability
and
Counting Rules
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
 Probability can be defined as the
chance of an event occurring.
SOME DEFINITIONS
 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.
 An event consists of outcomes.
SOME SAMPLE SPACES
Experiment
Toss a coin
Sample Space
S={Head
, Tail
}
Roll a die
Answer a true/false question
Toss two coins
S={True, False}
S={HH,
H HT, TH, TT}
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 4-3: GENDER OF CHILDREN
Find the sample space and tree diagram for the gender
of the children if a family has three children.
Use (B for boy) and (G for girl).
Solution :
B BBB
B
B
G
G BBG
B BGB
G
BGG
B GBB
B
G
G
G GBG
B GGB
G
GGG
 An event consists of outcomes of a probability
Experiment .
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
There are three basic interpretations of probability:

Classical probability

Empirical probability

Subjective probability
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
Probability Rules
There are four basic probability rules:

First Rule: 0 ≤ P(E) ≤ 1

Second Rule: If an event E cannot occur ,then the
probability is 0.
 Third
Rule: If an event E is certain , then the
probability of E is 1.
Fourth
Rule: ∑ p = 1
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.
Example 4-8: ROLLING A DIE
When a single die is rolled , find the probability of getting a 9 .
Solution :
Sample Space:
1,2,3,4,5,6
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 :
Sample Space:
1,2,3,4,5,6
Fourth Rule
P(no. less than 7)= 6/6 = 1
Note: This PowerPoint is only a summary and your main source should be the book.
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.
Example 4-10:
Find the complements of each event.
Event
Complement of Event
Rolling a die and getting a 4
Getting a 1, 2, 3, 5, or 6
Selecting a letter of the alphabet
and getting a vowel
Getting a consonant (assume y is a
consonant)
Selecting a month and getting a
month that begins with a J
Getting February, March, April, May,
August, September, October, November,
or December
Selecting a day of the week and
getting a weekday
Getting Saturday or Sunday
P (S)
P (E)
P (E)
Rule for Complementary Events
Example 4-11:
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.
Empirical Probability
Empirical probability(relative) relies on actual experience
to determine the likelihood of outcomes.
f frequency of desired class
PE  
n
Sum of all frequencies
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:
Type Frequency
A
22
B
5
AB
2
O
21
Total 50
a. A person has type O blood.
b. A person has type A or type B blood.
c. A person has neither type A nor
type O blood.
d. A person does not have type AB
blood.
Subjective probability
Subjective probability uses a probability value based on an
educated guess or estimate, employing opinions and inexact
information.
(based on: the person’s experience and evaluation of solution)
Examples: weather forecasting, predicting outcomes of sporting
events
*EX:
The probability that a bus will be in an accident on a specific run is
about 6% .
A ball is chosen at random from a box containing 5 black ,
8 red and 7 yellow balls . Find the probability that it is :
a) red
b) yellow
c) not black
Addition Rules for
Probability
*** Addition Rules ***
mutually exclusive
events
not mutually exclusive
events
if they cannot occur at
the same time (i.e., they
have no outcomes in
common)
Can occur at the same
time (i.e., outcomes in
common not mutually
exclusive)
P( A or B)  P( A)  P( B)
P(A∩B)= 0
P( A or B)  P( A)  P( B)  P( A and B)
** 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
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
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
1. Determine which events are mutually exclusive.
a) Select a student in your college: The student is in the second year and
the student is a math major.
b) Select a child: The child has black hair and the child has black eyes.
c) Roll a die: Get a number greater than 2 and get a multiple of 3.
d) Roll a die: Get a number greater than 3 and get a number less than 3.
Example 4-17:
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 :
Example 4-19:
Solution :
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.
Solution :
Staff
Nurses
Physicians
Females
7
3
Males
1
2
Total
8
5
Total
10
3
13
Multiplication Rules
*** Multiplication Rules ***
Independent Events
Dependent Events
If the fact that A occurs
does not affect the
probability of B
occurring.
The outcome of the first
event affects the
outcome of the second
event.
EX: Rolling a die and
getting a 6, then rolling
another die and getting 3
EX: Having high
grades and getting a
scholarship (‫)ثقافة‬.
P( A and B)  P( A).P( B)
P( A and B)  P( A).P( B / A)
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.
Example 4-27: MALE COLOR BLINDNESS
Approximately 9% of men have a type of color blindness ( ‫عمى‬
‫)ألوان‬that 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
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
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 ‫ )تأمين صاحب البيت‬with the
company .Of these clients ,27% also had automobile insurance
(A‫ )تأمين سيارة‬with 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 :
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
Solution :
Red
Box 1
Blue
Coin
Red
Box 2
Blue
** contains 20% defective transistors, contains 30% defective
transistors, and 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 . 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.
Conditional Probability
P  A and B 
P  B A 
P  A
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
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.
Find the probability that
a. the respondent answered yes, given that the respondent was a female.
b. the respondent was a male, given that the respondent answered no .
c. the respondent was a female or answered no.
PROBABILITIES FOR “ AT LEAST ”
EXAMPLE 4-36:

A coin is tossed 5 times . Find the probability
of getting at least 1 tail ?
E=at least 1 tail
E= no tail ( all heads)
P(E)=1-P(E)
P(at least 1 tail)=1- p(all heads)
1
)
5
2
1
31
 1 (
)
32
32
 1 (
** If 63% of children play computer games, and 4 of them
are chosen at random, find the probability that all four
play computer games.
A)0.158
A)2.52
A)1.48
A)0.019
** If 0.35 of men are smokers and 3 men are selected at
random, find the probability that at least one is a smoker.
a) 0.043
b) 0.274
c) 0.476
d) 0.725
Counting Rules
1- Fundamental Counting Rule
2- Permutation
3- Combination
1-Fundamental Counting Rule
 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
Event 1
k1
Event 1
k2
……………
……………
Event n
kn
EXAMPLE 4-39:
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?
# of
# of
# of
# of
colors types textures uses

 
 
7  2 
3
 2
84 different kinds of paint
If a menu has a choice of 8 appetizer (‫)مقبالت‬, 6 main courses
(‫)أطباق رئيسية‬, 5 deserts (‫)حلى‬, then the sample space for all
possible lunch can determined by using:
a) The addition rule.
b) The combination rule.
c) The fundamental counting rule.
-------------------------------------------------------------------------•The digits 0,1,2,3,4,5,6,7,8,9 are to be used in a four-digit ID
card. How many different cards are possible if repetitions are
permitted?
a) 16
b) 10000
c) 100000
•
2-Permutation
Permutation is an arrangement of n objects in a specific
order using r objects. (Order matters).

n!  n  n  1 n  2   n  r  1
n Pr 
 n  r !
r items
----------------------------------------------------------------- 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
Example 4-42:
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?
EXAMPLE 4-44:
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 :
8!
 336
8 P3 
5!
3-Combination
 Combination is a grouping of objects order does
not matter. (selection of distinct objects without regard to
order)
n!
n Cr 
 n  r  !r !
EXAMPLE 4-47:
How many combinations of 4 objects are there . Taken 2 at a
time?
Solution :
4c2
Example 4-49:
 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 :
7!
5!
Women: 7C3 
 35, Men: 5C2 
 10
4!3!
3!2!
EXAMPLE 4-51:
 A box contains 24 transistors , 4 of which are
defective . If 4 are sold at random ,
find the following probabilities :
a- Exactly 2 are defective .
P(exactly2aredefectives) 
c . C2 1140

10626
24 C4
4 2 20
b-Non is defective .
C4 4845
P(nodefectives ) 

10626
24 C 4
20
C- All are defective .
C4
1
P(alldefective) 

10626
24 C4
4
d-At least 1 is defective
P(atleast1defective)  1  P(nodefective)
C4
1615 1927
 1
 1

3542
3542
24 C 4
20
** In a statistics department, there are six teachers, 4 of which
are males. If 2 teachers are selected at random, what is the
probability that both of them are females?
a) 2/3
b) 2/5
c) 1/15
d) 1/3
** How many different ID cards can be made if there are 2 letters
followed by 2 digits and none of them can be used more than once?
a) 58500
how many outcomes are possible if both numbers selected must be
even?
a) 2
EXAMPLE 4-52:

A store has 6 TV Graphic magazines and 8 Newstime
magazines on the counter . If two customers purchased a
magazine, find the probability that one of each magazine
was purchased.
Solution:
C1.8 C1 6.8 48


P( 1 TV Graphic and 1 Newstime) 
91
91
14 C 2
6
* A new employee has an option of 5 health care
plans, 3 retirement plans and 2 different expense
accounts. Find the probability that a person can
select one of each option?
[5C1.3C1.2C1]/10C3=0.25
EXAMPLE 4-53:

A combination lock consist 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 .
Solution :
1
1
P( ABC )  3 
26
17576
EXAMPLE 4-54:

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 :
P(they are married to each other )=
8
1

8 .8
8
Revision
•1- A store
manager wants to display 6 different brands of
shampoo in a raw. How many different ways can this be done?
a) 120
b) 720
c) 6
d) 36
---------------------------------------------------------------------------------------------------------
2- If the probability that a person lives in a village is 0.6,
what is the probability that a person does not live in a village?
a) 0.2
b) 0.4
c) 0.3
d) 0.6
•
3- If P(A)=0.3, P(B)=0.4 and P(A and B)=0.10, then the
events A and B are said to be:
a) Mutually exclusive events.
b) Not mutually exclusive events.
c) Independent events.
d) Dependent events.
•
---------------------------------------------------------------------------------------------------------
4- If P(A)=0.3, P(B)=0.2 and P(A or B)=0.5, then the events
A and B are said to be:
a) Mutually exclusive events.
b) Not mutually exclusive events.
c) Independent events.
d) Dependent events.
•
5- A jar contains 3 red marbles, 7 green marbles and 10
white marbles. If a marble is drawn from the jar at
random, what is the probability that this marble is
white?
A-0.5
B-0.4
C-0.3
D-0.8
---------------------------------------------------------------------------------•
7- How many different letter arrangements can be made
from the word: “sample”?
A- 720
B- 820
C- 920
D- 1020
---------------------------------------------------------------------------------• 8- How many different letter arrangements can be made
from the word: “success”?
A- 720
B- 620
C- 520
D- 420
•
9- The sample space for the children gender in a family with
three children is (B: boy , G: girl):
a) 4
b) 8
c) S={BBB, BBG, BGB, BGG, GBB, GBG, GGB,GGG}
d) S={BBG, BGB, BGG, GBB, GBG, GGB}
•
--------------------------------------------------------------------------------------------------------
10- The statement “The probability that an earthquake will
occur in a certain area is 30%”. This is an example of:
a) Classical probability.
b) Empirical probability.
c) Subjective probability.
•
11- A store has 4 adventure stories (‫ )قصص مغامرات‬and 5
horror stories (‫ )قصص الرعب‬on the counter. If two customers
purchased a story, find the probability that one of each story
was purchased:
a) 5/9
b) 4/9
c) 50/100
•
--------------------------------------------------------------------------------------------------------
12- Which of these numbers cannot be a probability:
a) 0.01
b) 2%
c) -0.01
•
13- The probabilities of the events A and B are:
P(A and B)=0.2, P(B│A)=0.3. Find the P( A )
a) 0.3
b) 0.4
c) 0.5
•
--------------------------------------------------------------------------------------------------------
14- What type of probability uses sample space to determine
the probability that an event occur?
a) Classical probability.
b) Empirical probability.
c) Subjective probability.
•
15- A package contains 12 flash memories, 3 of which are
defective. If 4 are selected, find the probability of getting 3
defective flash memories?
a) 1/55
b) 5/55
c) 28/55
--------------------------------------------------------------------------• 16- How many different ways can a person select 3 cars from
6 cars in a specific order?
a) 120
b) 20
c) 66
•
•
17- Box A contains 4 red balls and 2 blue ball . Box B
contains 2 blue balls and 2 red ball .
A die is rolled .
If the outcome is an even number a ball is chosen at random
from Box A.
If the outcome is an odd number a ball is chosen at random
from Box B.
Find the probability of choosing a red ball.
a) 7/12
b) 1/12
c) 5/50
18- How many ways can a person select 6 physics books and 5
math books from 9 physics books and 11 math books?
9
C6 11C5
9
C6 5 C11
C6 /11C5
9 C6 11 C5
9
---------------------------------------------------------------------------
•
19-
Admin
TA
Total
Bachelor
7
8
15
Master
23
17
40
Total
30
25
55
•What
is the probability that a person is a Master and TA?
a) 0.309 b) 0.425 c) 1.5
•What
is the probability that a person is a Bachelor given that
he is a Admin?
a) 0.425 c) 0.233 c) 0.408
20- The probability of any event D is:
a) 0 ≤ P(D) ≤ 1
b) -1 < P(D) <1
c) 0 < P(D) ≤ 1
d) 0 ≤ P(D) < 2
--------------------------------------------------------------------------• 21- A die is rolled one time, find the probability of getting
number less than or equal 2 or an even number.
a) 1
b) 2/3
c) 5/3
d) 4/8
•
22- When an event is certain, what is its probability?
a) 1
b) 0
c) 0.5
--------------------------------------------------------------------------• 23- When an event is impossible, what is its probability?
a) 1
b) 0
c) 0.5
--------------------------------------------------------------------------• 24- a married couple has three children, find the probability
they are all boys?
a) 1/8
b) 3/8
•
25- The probability that a student has a car is 0.8, and the
probability that he has an I-Phone is 0.7, while the probability
that he either car or I-Phone is 0.6. Find the probability that he
has both.
a) 0.9
b) 0.6
c) 0.8
•
26The students in a class of 50 either take
Biology or Physics. There are 20 students who
take Physics, 15 male students who take
Biology and there are 23 are female students
in the class .Find the probability that a
student chosen at random is a male or takes
Biology?
.
𝟓𝟕 72
42
d. 27/55
a.
b. C.
𝟓𝟎
50
50
32- A student has to sell 2-book, from a collection of 6-math,
7-art, 4-economic books. How many choices are possible:
** Both books are to be the same subject:
a) 40
b) 42
** The books are to be on different subject:
a) 94
b) 90
•
•33
If a coin is tossed 4 times. Find the probability of getting at
least 1 head?
a) ½
b) 15/16
c) 1/8