Topic 5 Probability Permutation and Combination
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Transcript of Topic 5 Probability Permutation and Combination
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 35
PROBABILITY
Experiment : A situation involving chance or probability
that leads to result called outcomes.
Outcome : The result of single trial of an experiment.
Event : One or more outcomes of an experiment.
Probability : The measure of how likely an event is.
Problem 1 : A spinner has 4 equal sectors colored
yellow, blue, green and red.
a. What are the chances of landing on
blue after spinning the spinner?
b. What are the chances of landing on
red?
Experiment : Spinning the spinner.
Outcomes : The possible outcomes are landing on
yellow, blue, green or red.
Event : a. Landing on blue.
b. Landing on red.
Probability/Solution : a. The chances of landing on blue are 1
in 4, or one fourth. Therefore, the
probability of landing on blue is one
fourth.
b. The chances of landing on red are 1
in 4, or one fourth. Therefore, the
probability of landing on red is one
fourth.
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 36
Problem 2 : Toss a dice and observe the number that
appears on the upper face.
a. What is probability of rolling an even
number?
b. What is the probability of rolling a
number less than 4?
Experiment : __________________________________________
Outcomes : __________________________________________
Event : a. ____________________________________
b. ____________________________________
Probability/Solution : a. ____________________________________
____________________________________
b. ____________________________________
____________________________________
Simple event : An outcome or an event that cannot be
further broken down into simpler
components. Usually denote by E.
Sample space : Set of all simple events
In terms of simple events; sample space is a
collection of one or more simple events.
Usually denote by S.
Probability : If the probability space S consist of a finite
number of equal likely outcomes, then the
probability of an event E , written ( )P E is
defined as:
( )( )( )Sn
EnEP ==
Outcomes Possible of Number Total The
occur can event waysof number The E
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 37
Refer to Problem 1:
S = { Yellow (Y), Blue (B), Green (G), Red (R) }
Let Event A = { Landing on Blue } = { B }
Event B = { Landing on Red } = { R }
n (S) = 4; n (A) = 1; n (B) = 1
Therefore; a. P (A) = 4
1=
n(S)
n(A)
b. 4
1
n(S)
n(B)P(B) ==
Refer Problem 2:
S =
Let Event A =
Event B =
n(S) = ; n (A) = ; n (B) =
Therefore; a. P(A) =
b. P(B) =
TAKE NOTE:
The probability of an event A is a
number between 0 and 1 inclusive
( )0 1P A≤ ≤
EXAMPLE
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 38
Sample space, { }1,2,3,4,5,6,7,8,9,10S = , given A = { } { }1,3,5,7,9oddnumbers =
and B = { } { }2,4,6,8,10evennumbers = . Find ( ) ( )P A andP B
A card is drawn at random from an ordinary pack of 52 playing cards. Find
the probability that the card is a seven.
Two fair coins are tossed. Find the probability that the two heads are
obtained.
A ball is drawn from a box containing 10 red, 15 black, 5 green and 10 yellow
balls. Find the probability that the ball is
a. Black
b. Not green or yellow
c. Not yellow
d. Red or black or green
e. Not blue
EXAMPLE:
EXAMPLE:
EXAMPLE
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 39
Venn diagram : A simple way of illustrating the relationship
between sets. The sets are represented by a
simple plane area, usually bounded by a
circle or a closed space.
The data below shows a survey of 305 college students.
125 take Mathematics
115 take Accounting
110 take Science
35 take Mathematics and Accounting
30 take Mathematics and Science
34 take Accountings and Science
10 take Mathematics, Science and Accounting
a. Illustrates the information using a Venn Diagram
b. How many take Mathematics only
c. How many takes Accounting but not Science
d. How many take Science or Accounting
e. How many students who are not taking any subject
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 40
Tree diagram : Illustrates experiments that can be
generated in stages. Each level of
branching on the tree corresponds to a
step required to generate the final
outcome.
A medical technician records a person’s blood type and Rh factor. List the
sample events in the experiment.
Two balls are drawn from a box containing 10 red, 15 black, 5 green and 10
yellow balls. If the balls are selected
i) First with replacement and again without replacement.
ii) First without replacement and again without replacement.
Find the probability if the balls selected are:
a. Of the same color
b. Not the same colors
c. Green and yellow
d. Both black
EXAMPLE:
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 41
Probability table : A tabulated form to illustrate experiments
that can be generated in stages.
A medical technician records a person’s blood type and Rh factor. List the
sample events in the experiment.
Probability Table:
Blood Type
Rh Factor A B AB O
Negative
Positive
Refer to the table illustrated below.
Length of Service
Loyalty < 1 year 1-5 years 6-10 years >10 years Total
Remain 10 30 5 75 120
Not Remain 25 15 10 30 80
Total 35 45 15 105 200
a. What is the probability of selecting at random an executive who has
been working for more than 10 years?
b. What is the probability of selecting at random an executive who would
not remain with the company and has less than one year of service?
EXAMPLE:
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 42
Operations of set theory:
i. Unions A B∪ : The union of events A and B, is the event
that A or B or both occur.
ii. Intersection A B∩ : The intersection of events A and B, is the
event that both A and B occur.
iii. Complements A′ : Also denoted by CA and A represents the
area (values) not in A
iv. Disjoint 0A B∩ = : Also called mutually exclusive events are
two events that does not contain have
similarities.
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 43
Two fair coins are tossed, and the outcome is recorded. These are the events
of interest:
A: Observe at least one head
B: Observe at least one tail
Define the events A, B, A ∩ B, and A ∪ B as collections of simple events, and
find their probabilities.
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 44
An experiment consists of tossing a single dice and observing the number of
dots that shows on the upper face. Events A, B and C are defined as follows:
A : Observe a number less than 4
B : Observe a number less than or equal to 2
C : Observe a number greater than 3
Draw a Venn diagram that illustrates the situation:
Find the following probabilities:
1. P( S )
3. P( B )
4. P(A ∩ B ∩ C)
5. P(A ∩ B)
6. P(A ∩ C)
7. P(B ∩ C)
8. P(A ∪ C)
9. P(B ∪ C)
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 45
Addition Rule:
The addition rule helps solve probability problems that involve two events.
When asked to find the probability of A or B, mean that A can happen, or B
can happen, or both can happen together. This is what is stated in the
addition rule.
Notation for Addition Rule:
P (A or B) = P (in a single trial, event A occurs or event B occurs or
they both occur)
Additive Rule of
Probability : Deals with unions of events
Given two events, A and B, the probability of their union, A ∪ B, is equal to
* Mutually exclusive : Two events are mutually exclusive when
one event occurs the other cannot.
When events A and B are mutually
exclusive, P( A ∩ B) = 0
Suppose we roll two dice and want to find the probability of rolling a sum of 6
or 8. This can be written in words as P(6 or 8) or more mathematically is
P(6 8). So what is the probability of getting a 6 or an 8 or both?
P (6) = 5/36
P (8) = 5/36
P (6 and 8 together) is impossible so the probability is 0.
So P(6 8) = 5/36 + 5/36 - 0 = 10/36 = 5/18
For the additive rule: P (A ∪ B) = P (A) + P(B) – P (A ∩ B)
If A and B are mutually exclusive, then P (A ∩ B) = 0
thus P (A ∪ B) = P (A) + P (B)
P (A ∪ B) = P (A) + P(B) – P (A ∩ B)
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 46
1. You are going to pull one card out of a deck. Find P(Ace King).
2. You are going to roll two dice. Find P(sum that is even or sum that is a
multiple of 3).
3. Drawing a card from an ordinary deck of cards. Find P(three or jack),
P(three or jack), and P(club or four).
Remember that:
- OR (the union symbol ) means that one
or the other or both events can happen. - AND (the intersection symbol ∩ ) means
that two events happen.
EXAMPLE
CHAPTER 4 - PROBABILITY
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Multiplication Rule:
The multiplication rule also deals with two events, but in these problems the
events occur as a result of more than one task (rolling one die then another,
drawing two cards, spinning a spinner twice, pulling two marbles out of a
bag, etc).
When asked to find the probability of A and B, we want to find out the
probability of events A and B happening.
Notation for Multiplication Rule:
P (A and B) = P (event A occurs in a first trial and event B occurs in a
second trial / event A and event B occur together)
Multiplicative Rule of
Probability : Deals with intersections of events
Given two events, A and B, the probability
that both of two events occur is
* Independent events : Two events A and B are said to be
Independent if and only if either
( ) ( )|P A B P A= or ( ) ( )|P B A P B=
Otherwise, the events are said to be
dependent.
For multiplicative rule: P (A ∩ B) = P (A) P (B|A)
@ P (A ∩ B) = P (B) P (A|B)
If A and B are independent, then P (A ∩ B) = P (A) P (B)
Similarly, if A, B and C are independent events, then the probability
that A, B, and C occur is P (A ∩ B ∩ C) = P (A) P (B) P (C)
P (A ∩ B) = P (A) P (B|A)
@
P (A ∩ B) = P (B) P (A|B)
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 48
1. Suppose we roll one die followed by another and want to find the
probability of rolling a 4 on the first die and rolling an even number on
the second die.
P (4) = 1/6
P (even) = 3/6
So P (4 even) = (1/6) (3/6) = 3/36 = 1/12
� While the rule can be applied regardless of dependence or
independence of events, we should note here that rolling a 4 on
one die followed by rolling an even number on the second die
are independent events.
� Each die is treated as a separate thing and what happens on
the first die does not influence or effect what happens on the
second die. This is our basic definition of independent events:
the outcome of one event does not influence or affect the
outcome of another event.
Notice:
In this problem we are not dealing with the sum of both dice. We are
only dealing with the probability of 4 on one die only and then, as a
separate event, the probability of an even number on one dies only.
2. Suppose you have a box with 3 blue marbles, 2 red marbles, and
4 yellow marbles. We are going to pull out the first marble, leave it out,
and then pull out another marble. What is the probability of pulling out
a red marble followed by a blue marble?
� We can still use the multiplication rule which says we need to
find P(red) P(blue). But be aware that in this case when we go
to pull out the second marble, there will only be 8 marbles left in
the bag.
P (red) = 2/9 P (blue) = 3/8
P (red blue) = (2/9)(3/8) = 6/72 = 1/12
Notice:
The events in this example were dependent. When the first marble was
pulled out and kept out, it affected the probability of the second
event. This is what is meant by dependent events.
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 49
1. There are 11 marbles in a bag. Two are yellow, five are pink and four
are green. Suppose you pull out one marble, record its color, put it
back in the bag and then pull out another marble. Find
a. P (yellow and pink)
b. P (pink and green)
2. Suppose you are going to draw two cards from a standard deck. What
is the probability that the first card is an ace and the second card is a
jack.
Conditional Probability:
This rule is applied when you have two events and you already know the
outcome of one of the events.
Conditional Probability : Probability obtained with the additional
information that some other event has
already occurred.
The conditional probability of B, given that A has occurred, is:
The conditional probability of A, given that B has occurred, is:
( )( | ) ( ) 0
( )
P A BP B A if P A
P A
∩= ≠
( )( | ) ( ) 0
( )
P A BP A B if P B
P B
∩= ≠
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 50
1. A math teacher gave her class two tests. 25% of the class passed both
tests and 42% of the class passed the first test. What percent of those
who passed the first test also passed the second test?
� This problem describes a conditional probability since it asks us
to find the probability that the second test was passed given
that the first test was passed.
( )( )
( )
%606.042.0
25.0===
=FirstP
SecondandFirstPFirst|SecondP
2. A survey of 500 adults asked about college expenses. The survey asked
questions about whether or not the person had a child in college and
about the cost of attending college. Results are shown in the table
below.
Cost Too Much Cost Just Right Cost Too Low
Child in College 0.30 0.13 0.01
Child not in College 0.20 0.25 0.11
Suppose one person is chosen at random. Given that the person has a
child in college, what is the probability that he or she ranks the cost of
attending college as “cost too much”?
This problem reads:
P (cost too much | child in college) or
P (cost too much given that there is a child in college)
According to the conditional probability rule:
P(cost too much child in college) =
P(cost too much child in college) = 0.30
P(child in college) = 0.30 + 0.13 + 0.01 = 0.44
Therefore;
EXAMPLE:
CHAPTER 4 - PROBABILITY
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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 59
=
35
Complementation : The complement of an event A, denoted
by CA , consists of all the simple events in
the sample space S that are not in A.
1. For events A and B it is known that
( ) ( ) ( )2 7 5,
3 12 12P A P A B andP A B= ∪ = ∩ = . Find
a. ( )P B
b. P( B|A )
2. For events A and B it is known that ( ) ( ) ( ) 0.1P A P B andP A B= ∩ = and
( ) 0.7P A B∪ = . Find ( )P A .
3. Given ( ) 4.0=AP and ( ) 5.0=BP
a. If A and B are mutually exclusive events, find P (both A and B)
b. If A and B are independent events, find P (either A and B)
( ) ( )' 1P A P A= −
EXAMPLE:
36
4. In a color preference experiment, 8 toys are placed in a container. The
toys are identical except for color where 2 are red, and six are green.
A child is asked to choose two toys at random, once with replacement
and once without replacement.
What is the probability that the child chooses the two red toys for both
ways? Draw tree diagram for each method to illustrate your working.
(Relate the tree diagram using the multiplicative rule of probability)
5. An experiment consists of tossing a single dice and observing the
number of dots that shows on the upper face. Events A, B and C are
defined as follows:
A : Observe a number less than 4
B : Observe a number less than or equal to 2
C : Observe a number greater than 3
Are events A and B independent? Are the events A and B mutually
exclusive?
37
6. Toss two coins and observe the outcome. Define these events:
A: Head on the first coin.
B: Tail on the second coin.
Are events A and B independent?
TAKE NOTE:
For this type of question,
ensure you prove the steps
accordingly.
38
Additional Rules using complements of events:
P (A ∪ B )’ = P ( A’ ∩ B’ )
P (A ∩ B )’ = P ( A’ ∪ B’)
P ( A ∩ B’ )
P ( A’ ∩ B )
P ( A’ )
39
Given ( ) 0.59P A = , ( ) 0.30P B = and ( ) 0.21P A B∩ = , find
a. ( )P A B∪
b. ( )' 'P A B∪
c. ( )'P A B∩
d. ( )' 'P A B∩
EXAMPLE:
40
Given ( ) 0.40P A = , ( ) 0.70P B = and ( ) 0.2P A B∩ = , find
a. ( )/P A B
b. ( )'/P A B
c. ( )/ 'P A B
d. ( )'/ 'P A B
EXAMPLE:
41
Bayes’ Theorem : The probability of event A, given that event B has subsequently occurred, is
( )( ) ( )
( ) ( )[ ] ( ) ( )[ ]A|BPAPA|BPAP
A|BPAPB|AP
+=
Suppose 5% of the population of Umen (a fictional country) have a disease
that is peculiar to that country. Consider there is technique to detect the
disease although it is not very accurate. The probability that the test indicates
disease is present is 0.90. The probability that a person actually does not have
the disease but test indicates the disease is present is written as 0.15.
Illustration using tree diagram:
Define the situation
Let D : Have the disease N : Does not have the disease
X : Test show the disease is present (positive)
X : Test shows the disease is not present (negative)
Draw tree diagram using the terms defined above
EXAMPLE:
42
Possible questions:
a. What is the probability of the patient is tested positive?
b. What is the probability of the patient is tested negative?
c. What is the probability that the patient has the disease and tested positive?
d. What is the probability that a person has the disease, given that he or she is
tested positive?
e. What is the probability that a person has the disease, given that he or she is
tested negative?
43
When a person needs a taxi, it is hired from one of the three firms : X, Y and Z.
Of the hiring, 40% are from X, 50% are from Y and 10% are from Z. For taxi hired
from X, 9% arrived late. The corresponding percentages for taxis hired that
arrived late from firm Y and Z are 6% and 20% respectively.
a. Construct a tree diagram for the above information.
b. Calculate the probability that the next taxi hired
i. will be from X and will not arrive late
ii. will arrive late
c. Given that a call is made for a taxi and that it arrives late, find the
probability that it came from Y.
44
Permutations : The number of ways we can arrange n
distinct objects, taking them r at a time is
!
( )!
n
r
n
n rP =−
Permutation is an ordered arrangement of all or part of a set o items.
The 3 letters, P, Q and R can be arranged in the following ways
PQR PRQ QPR RPQ RQP QRP
Each of the arrangement is called permutation of the letters P, Q and R.
Using the formula for permutation, the number of permutations of the 3 letters
taken 3 at a time is
( )3
3
3!
3 3 !
3!
0!
3 2 1
6
P =−
=
= × ×
=
Similarly, if the question requires the number of permutations of 3 letters taken
2 at a time is
( )3
2
3!
3 2 !
3!
1!
3 2 1
6
P =−
=
= × ×
=
The permutations are PQ, PR, QP, QR, RP, RQ.
.
TAKE NOTE:
PR and RP are considered as
2 different permutations
45
The numbers of permutations of n items with p are alike of a first kind, q is alike
of a second kind, r is alike of a third kind, and so on is
!
! ! !.....
n
p q r
The number of permutations if the n items are arrange in a circle
( )1 !
! ! !.....
n
p q r
−
Calculate the number of permutations that can be formed using letters from
the word STATISTICS.
3 lottery tickets are drawn from a total of 50. If the tickets are distributed to
each of 3 employees in the order in which they are drawn, the order will be
important. How many simple events are associated with the experiment?
46
In how many ways can the letters A, B, C, D and E can be arranged without
repetition when
a. all the 5 letters are taken at a time
b. 4 of the letters are taken at a time
c. 3 of the letters are taken at a time
47
Find the number of arrangements of the word MATHEMATICS and THEMSELVES
How many 3-letter words can be formed from the letters in the word
ABSOLUTE?
How many of these 3-letters words
a. Contain the letter S,
b. Do not contain any vowel
48
Combinations : The number of distinct combinations of n
distinct objects that can be formed, taking
them r at a time is
)!(!
!
rnr
n
Cn
r −=
Combination of a set of items is a selection of one or more of the items with
no consideration given to the order or arrangement of the items
The number of combinations of 3 letters, P, Q and R, taken 2 at a time is
( )3
2
3!
2! 3 2 !
3 2 1
2 1 1
3
C =−
× ×=
× ×
=
The combinations are: PQ or QP, QR or RQ and PR or RP.
The relation between permutation and combination can be written as
!r
PC
n
rn
r=
TAKE NOTE:
PR and RP are considered
the same combination
49
A printed circuit board may be purchased from 5 suppliers. In how many
ways can 3 suppliers be chosen from the 5.
A committee is to be formed from 8 men and 4 women. Find the number of
ways this committee can be formed consisting of
a. 7 members
b. 7 members, 5 men and 2 women
c. 7 members, men more than the women
EXAMPLE:
EXAMPLE:
TAKE NOTE:
Permutations and
Combinations techniques
are easier to perform by
using the box method
50
1. One card will be randomly selected from a standard 52-card deck.
What is the probability the card will be queen?
2. The National Center for Health Statistics reports that of 883 deaths, 24
resulted from an automobile accident, 182 from cancer and 333 from
heart disease. What is the probability that a particular death is due to
an automobile accident?
3. An ordinary dice is thrown. Find the probability that the number
obtained
a. Is a multiple of 3
b. Is less than 7
c. Is a factor of 6
4. If { }1,2,3,4,5,6,7,8,9S = , { }2,4,7,9A = , { }1,3,5,7,9B = , { }2,3,4,5C = , and
{ }1,6,7D = . List the elements of the sets and the probabilities
corresponding to the following events:
a. CAc ∪
b. c
CB ∩
c. ( )cc
BS ∩
d. ( ) BDCc ∪∩
e. ( )c
BA ∩
5. If 3 coins are to be thrown simultaneously together, list down all the
elements.
6. Referring to question 5, what is the probability of obtaining
a. One head
b. At least two heads
c. All tails
EXERCISE:
51
7. A trainee has conducted a survey on the hand phone market among
students in private higher educational institutes (IPTS) around Kuala
Lumpur. From the survey among 130 students, the following data were
gathered:
53 students use MOTOROLA phones
58 students use NOKIA phones
54 students use SONY ERRICSON phones
30 students use NOKIA phones only
25 students use SONY ERRICSON phones only
28 students use MOTOROLA phones only
8 students use all three phones
a. Present the information gathered using a Venn Diagram
b. How many students use only MOTOROLA and NOKIA
phones?
c. How many students use at least 2 hand phones?
d. How many students did not use any off the phones
above?
8. The probability of surviving a certain transplant operation is 0.55. If a
patient survives the operation, the probability that his or her body will
reject the transplant within a month is 0.20. What is the probability of
surviving both of these critical stages?
9. In a college graduating class of 100 students, 54 studied mathematics,
69 studied history and 35 studied both mathematics and history.
a. Draw a Venn diagram that illustrates the above situation
b. Using the diagram drawn in (a), find the probability of
i. The student takes mathematics and history
ii. The student does not take either of these subjects
iii. The student takes history but not mathematics
52
10. A selected group of employees of Samsung Manufacturing is to be
surveyed about a new pension plan. In-depth interviews are to be
conducted with each employee selected in the sample. The
employees are classified as follows:
Classification Event No of
employees
Supervisors A 120
Maintenance B 50
Production C 1460
Management D 302
Secretarial E 68
What is the probability that the first person selected is
a. Either in maintenance or secretarial
b. Not in management
11. There are 100 students in a first year college intake, 45 are male
students of which 36 are studying programming. From the total female
students, 42 are studying programming. Draw a probability table to
illustrate the situation. What is the probability that a student selected
randomly is studying programming knowing that the student is a male?
12. A coin is loaded so that the probabilities of heads and tails are 0.52
and 0.48 respectively. If the coin is tossed three times, what is the
probabilities of getting
a. All heads
b. Two tails and a head in that order
13. Given that A and B are two events with probabilities P(A) = 0.4, P(A/B)
= 0.2 and P(B) = 0.15. Find
a. P(B/A)
b. P(A ∪ B)
14. Two event R and T are defined in a sample space with probabilities
P(R) = 0.35, P(T) = 0.28 and P(RT) = 0.06. Find
53
a. P(R ∪ T)
b. P(RT’)
c. P(R’T’)
15. Two ordinary dice are thrown. Find the probability that the sum of the
scores obtained
a. is a multiple of 5
b. is greater than 9
c. is a multiple of 5 or is greater than 9
d. is a multiple of 5 and is greater than 9
16. Two events A and B are such that P(A) = 0.3 and P(B) = 0.4 and P(A/B)
= 0.1. Calculate the probabilities
a. that both the events occur
b. at least one of the event occur
c. B occur, given that A has occurred.
17. Three people in an office decide to enter a marathon race. The
respective probabilities that they will complete the marathon are 0.9,
0.7, and 0.6. Find the probability that at least one will not complete the
marathon. Assume that of each is independent of the performances of
the others.
18. Suppose A and B are two event with P(B) = 0.5, P(A / B) = 0.4 and P(A’ /
B’) = 0.3. Find
a. P(A ∩ B)
b. P(A ∪ B)
19. Let S and T are two events such that P(S) = 0.6, P(T) = 0.5 and P(S’ ∩ T) =
0.3. Obtain
a. P(S ∩ T)
b. P(T / S’)
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20. Suppose that A and B are two events with P(A) = 1/5, P(B) = 1/4 and P(A ∪ B) = 1/3. Find
a. P(A ∩ B)
b. P(A / B)
c. P(B’ / A’)
21. A and B are two events where P(B) = 1/6 , P(A ∩ B) = 1/12, and P(B / A)
= 1/3. Find
a. P(A)
b. P(A / B’)
22. Given that P(A) = 0.8, P(A / B) = 0.8 and P(A ∩ B) = 0.5. Find
a. P(B)
b. P(B / A)
c. P(A ∪ B)
23. A sharpshooter hits a target with probability 0.75. Assuming
independence, find the probability of getting
a. A hit followed by two misses
b. Two hits and a miss in any order
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24. A box contains 5 red bulbs, 4 blue bulbs and 3 yellow bulbs. Three balls
are selected at random from the box.
a. Find the probability that all three bulbs are of the same color.
b. Find the probability that the three bulbs are of different colors if
it is known that one of them is yellow.
25. A box contains 15 mathematics books and 10 music books. Two books
are selected one at a time without replacement, find
a. The probability that the first book selected is a mathematics
book
b. The probability that the second book selected is a music book if
the first book selected is a mathematics book
26. Given that P(C ∩ D) = ¼, P(C/D) = 1/3 and P(D/C) = 3/5.Find
a. P(C)
b. P(D)
c. P(C / D’)
27. If A and B are mutually exclusive events with ( ) 0.5P A = and ( ) 0.3P B = ,
find
a. ( )'P A
b. ( )'P B
c. ( )P A B∪
d. ( )' 'P A B∪
e. ( )'P A B∩
f. ( )' 'P A B∩
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28. Given the following probabilities ( ) ( ) 95.0,3.0 =∪= BAPAP
a. Find ( )BP if A and B are mutually exclusive
b. Find ( )BP if A and B are independent events
29. A box contains 4 red balls, 5 green balls and 8 blue balls
a. A ball is drawn at random from a box. Find the probability that it
is
i. Red
ii. Green or blue
b. Three balls are drawn one by one from the box. Find the
probability that the balls are drawn in the order of red, green
and blue if the ball is
i. Replaced
ii. Not replaced
30. From a box containing 6 black beads and 4 green beads. 3 beads are
drawn in succession, each beads being replaced in the box before the
next draw is made. Draw a tree diagram that represents the above
experiment. Find the probability that
a. All three are of the same color
b. Each color is represented
31. If the probability that student A will fail a certain statistics examination is
0.5, the probability that student B will fail the examination is 0.2 and the
probability that both student A and student B will fail the examination is
0.1, what is the probability that:
a. At least one of the two students will fail the examination?
b. Neither student A nor student B will fail the examination?
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32. Routine physical examinations are conducted annually as part of a
health service program for a company for its employees. It was
discovered that 8% of the employees need corrective shoes, 15% need
dental work and 3% need both health service. What is the probability
that an employee selected at random will need either corrective
shoes or major dental work?
33. An automatic Shaw machine inserts mixed vegetables into a plastic
bag. Experience revealed some packages were underweight and
some overweight, but mostly had satisfactory weight.
Weight Probability No of packages
Underweight, A 0.025 100
Satisfactory, B 0.900 3600
Overweight, C 0.075 300
1.000 4000
a. What is the probability of selecting three packages from the
production line and finding all three are underweight?
b. What does this probability mean?
34. The probability that a regularly schedule flight departs on time is 83.0)( =DP , the probability that it arrives on time is 92.0)( =AP , and
the probability that it departs and arrive on time is 78.0)( =∩ ADP .
Find the probability that a plane
a. Arrives on time given that it departed on time
b. Departed on time given that it has arrived on time
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35. A firm uses 3 local hotels to provide accommodations for its clients.
From past experience, it is known that 15% of its clients are assigned
rooms at Hotel A, 55% at Hotel B and 30% at Hotel C. It is known that
the probabilities Hotel A, Hotel B and Hotel C will be facing water
problems are 0.03, 0.02 and 0.07 respectively.
a. Draw a tree diagram for the above data.
b. What the probability that a client will be assigned a room
without water problems?
36. The Credit Department of a supermarket reported that 75% of their
sales are in cash while the remaining 25% are using credit cards. 80% of
the cash sales and 30% of the sales using credit cards are for the sales
amount of less than RM100.
a. Draw a tree diagram for the above problem.
b. What is the probability of a sale less than RM100?
c. Puan Rahimah has just bought a calculator that cost less than
RM100, what is the probability that she paid cash?
37. The number of employees of store A, B and C are distributed in the
percentage 20%, 30% and 50%. 50%, 60% and 70% of these are women
respectively. One employee is selected at random.
a. Draw a tree diagram to summarize the above problem.
b. What is the probability that the employee is a male?
c. Given that the employee is a male, what is the probability that
he works in store C?
38. In an annual sport election, there are three possible candidates. The
probability of Mr. Ahmad, Mr. Yaman and Mr. Zain being nominated
are 0.25, 0.45 and 0.3 respectively. If the nominated candidates are
taking part in the election, the probability that the election is won by
Mr. Ahmad, Mr. Yaman and Mr. Zain are 0.6, 0.35 and 0.60 respectively
a. Draw a probability tree to represent the above situation.
b. Given that someone won the election, what is the probability
that he is Mr. Zain?
39. The Snapquick Store gets its supply of camera from three suppliers A, B
and C in the ratio of 5 : 3 : 2. However, some of the cameras supplied
59
are faulty. 10% of the cameras obtained from supplier A are faulty, so
are 5% from supplier B and 3% from supplier C.
a. Construct a tree diagram.
b. One camera is chosen at random from the store. What is the
probability that the camera
i. is faulty
ii. was obtained from supplier A if it is found to be faulty.
40. Of a group of students studying at a School of Management, 48% are
male and 52% are female. 20% of the males and 30% of the females
from this group, major in Marketing. Find the probability that:
a. A student selected at random from this group is a female
majoring in Marketing.
b. A student selected at random from this group is not majoring in
Marketing
c. A marketing student selected at random from this group is a
male.
41. In how many ways can the letters A, B, C, D, E and F can be arranged
without repetition when
a. All the 6 letters are taken at a time
b. 5 of the letters are taken at a time
c. 3 of the letters are taken at a time
42. Find the number of arrangements of the following words
a. CALENDAR
b. MALAYSIA
c. CALCULATOR
d. PROBABILITIES
43. Suppose repetitions are not permitted
a. How many three digit numbers can be formed from the six digits
2, 3, 4, 5, 7 and 9?
60
b. How many of these numbers are less than 400?
c. How many are even?
d. How many are odd?
e. How many are multiples of 5?
44. Solve the problem in question 43, if repetitions are permitted.
45. { }: 2,4,6,7A . How many 4 digits numbers can possibly be formed from
set A if
a. each digit can be used only once
b. each digit can be used more than once
c. 4th digit is odd number
46. In how many ways can the first, second and third prizes are awarded in
a class of 30?
47. If there are 8 vacant seats in a bus, in how many ways can 5 persons
seat themselves?
48. In how many ways can 7 persons sit at a round table?
49. How many 4-digit numbers can be formed by using the digits 0 to 9 if
a. Repetition is allowed
b. Repetition is not allowed
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50. Using the word MISSISSIPPI, find the number of permutations that can
be formed
a. From all the letters of the word
b. If the words are to begin with the letter I
c. If the words are to begin and end with the letter S
d. If the two P’s are to be next to each other
e. If the four S’s are to be next to each other
51. A class consists of seven men and five women
a. Find the number of committees of five that can be formed from
the class
b. Find the number of committees that consist of three men and
two women
c. Find the number of committees that consist of at least one man
and at least one woman
52. A bag contains five red marbles and six white marbles.
a. Find the number of ways that four marbles can be drawn from
the bag
b. Find the number of ways that four marbles can be drawn from
the bag, if two of the marbles must be red and two of the
marbles must be white
c. Find the number of ways that four marbles can be drawn from
the bag if the four marbles must be of the same color
53. How many ways can the letters of the word ‘DIGIT’ can be arranged?
How many of these arrangement
a. The I’s are together?
b. The I’s are separated?
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54. A committee consist of 4 members will be selected from 5 polices, 2
lawyers and 3 doctors. How many ways the committee can be formed
if
a. The committee will consist of 3 polices
b. The committee will consist of no more than a lawyer
55. A delegation of 4 members is to be formed from 7 chemistry lecturers
and 5 biology lecturers. In how many ways can the group be formed if
at least 2 chemistry lecturers should be in the group?
56. There are 7 women and 8 men in a committee. In how many ways can
a group of 3 people be selected from the committee if the 3 should be
of all men or all women?
57. In how many ways can a president, an assistant president, a treasurer
and a secretary be chosen from 8 selected members of an
organization?
58. Students of Diploma in Computer Science consist of 10 males and 15
females. What is the probability of selecting a committee of 5 males
and four females if Siti must be one of them?
59. The eleven letters of the word BOOKSHELVES are arranged in a line.
a. How many distinct arrangements can be done?
b. If an arrangement is chosen at random, what is the probability
that the two O’s are together?
60. In an examination a student has to answer 6 out of 10 questions.
a. How many choices does the student have?
b. How many choices does the student have if he/she must answer
the first two questions?
c. How many choices does the student have if he/she must answer
at least 3 of the first four questions?