Notesale: Turn your study into money

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CHAPTER ONE

Objectives
At the end of this chapter, students will be able to:
i) Know the several schools of thoughts and their contribution to probability
...

INTRODUCTION TO PROBABILITY
Life would have not been safe and enjoyable supposing we lacked the ability to predict it
...
Several tools, especially mathematical
tool have made it possible for the prediction of certain events in other to plan wisely against the
future
...

The word “probability” also referred to as “chance” is mostly used in our day to day activity
either knowingly or unknowingly, for example, we may come across statements like “ There is
a probability that the sun might shine today” or “there is a probability that it might rain for the
next two days”
...
Probability is the relative
frequency of an event which ranges from zero to one (0-1), i
...
zero for an event that do not
occur, while one for an event certain to occur
...
It
should be noted that the task of assigning the numbers should depend on the interpretation
...

Their approaches to probability are:
1)
2)
3)
4)

Classical or prior approach to probability
Relative frequency theory to probability
Axiomatic approach to probability
Subjective approach to probability

CLASSICAL APPROACH TO PROBABILITY
This approach to probability originated in the 18th century from problems related to games of
chance, such games that involve throwing of dices, cards, coins or desk cards
...
One of the proponents is a
French Mathematician Laplace
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Modern terminology is a term that describes the process which results to different possible
outcomes or observation
...
These conditions are:
i)
ii)
iii)

Are they equally likely?
Do the events form an exhaustive set?
Are the events mutually exclusive?

For an event to be “equally likely” each outcome of an experiment must have the same chance
of appearing as any other, e
...
a die with an occurrence, 1,2,3,4,5,6 are equally likely events to
occur
...
It is mathematically expressed as:
( )
We should note the following while calculating probability:
i)
Total number of equal likely cases
ii)
Number of Favourable case
( )
The above expression, explains a situation whereby, “A” might occur in a way, given the
total number of observation, “n”
...
But in certain
cases were we have exhausive event, the probability of A, not occuring A! is given as:
( )

( )

( )

since the sum of the events in exhausive case(successful and unsuccessful) is equal to one
...
+an = 1
( )

( )

; where

A = success and
B = failure
Illustration; A basket containing four red balls, three blue balls, and six orange balls was
placed on a table for experiment on probability
...

Solution: ( )
where;
= 4 ,
therefore ( )

,

The probability of taking a red ball is


...
This theory tries to solve the problem that the classical
school of thought usually encounter, when it comes to frequency distribution
...
They were of the opinion that the accuracy of the probability (P(A)) is
the function of the number of events/observation(n), as “n” approaches infinity( ), the relative
frequency tends to approach a limiting value of mathmatically expressed as
( )

( )

( )

The expression states that the probability is the value that tends to as n approaches infinity
...
However, the more number of time we carry out same experiment, the more
accurate the result will become
...
5
respectively which indicate 50% of each events
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For this reason, modern probability is an undefined concept the same(as ) way point and
line are undefined in geometry
...
N
...
This approach explains that probability have no precise definition, rather they give
axioms or postulates on which probability calculations are based
...
There are three axioms that govern the field of probability theory for
finite sample spaces
...
If the event cannot take place, its
probability shall be zero, and if it is certain to occur, its probability shall be one (1)
...
e
...

Iii) if A and B are mutually exclusively (or disjoint) events, then the probability of occurance
of either A or B denoted by P(
) shall be given by
P(

)

( )

( )

SUBJECTIVE APPROACH TO PROBABILITY
This approach to probability was first introduced in a book tittled “Foundation of Mathmatics”
written by Frank Ramsey in 1926
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This approach to a large extent is influenced by differences in
experience, values and attitudes
...
The
subjective approach sees probability as assigned to an event by an individual based on the
evidence available
...
For example, a lecturer can only state the
probability of his student passing his semester examination, after judging from his previous
performance in the continous asessments which may involve tests and assignments
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In conclusion, the four explained approaches to probability above are important in solving
problems related to probability
...

Experts still argue on the most appropriate theory to be adopted
...

The word “mathematical” is an adjective that qualifies or describes the noun “probability”
...
It seeks
to use mathematics as a tool to solve most problems that have been the drawbacks of
probability
...

Random event may only occur when the experiment is repeated as many times as possible
without we having control over the possible outcomes
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MUTUALLY EXCLUSIVE EVENTS
Supposing the following events, E1, E2, E3 occur in a given experiment on probability, these
events are said to be mutually exclusive if they do not occur simultaneously
...
Take for instance, if a die is tossed
once, there are six events, 1,2,3,4,5,6, which may likely show up
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Before we consider problems of probability related to mutually exclusive
events, let us consider the basic theorems that govern mutually exclusive events
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( )
Let P(A) be probability of Apples and
P(B) be probability of Mangoes
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This is represented by (
(

( )

)

( )

( )

( )

Example 1: what is the probability of picking oranges and mango, from the
previous example
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ADDITION-SUBTRATION THEOREM: When events are not mutually exclusive, i
...
there
is a probability that those events can both occur, addition rule will be modified
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+

EXHAUSTIVE EVENTS
Different set of events E1,E2 E3, E 4……En are said to be exhaustive events if their totality
include all the possible outcomes of a random experiment(sample space)
...
For events to be exhausive, the sumation of their probabilities must be
equal to a unity (1)
...

Example

1:

supposing

the

probability

of

( )
Where

hitting

a

target

with

an

arrow

is

( )

( )

,
( )
( )

( )

( )
The above solution interpretes that P(B) is the probability of the event A not occuring and it is
expressed as A!
...

A or C is given as (

)

( )

( )

B or C is given as (

)

( )

and that of (B or C) is

( )

( )
( )

( )
( (

( )
))

then calculate

( )

( )

( )

( )

( )

( )
Therefore P(C) is given as
CONDITIONAL EVENTS
For events to be conditional, there must exist a dependent relationship between them
...
Conditional probability is given by P(A/B) where event A depends on
the occurance of event B or P(B/A) when the reverse is the case
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The general multiplication rule in its modified form in terms of
conditional probability applies
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Example : A basket contains 5 blue toys and 7 green toys
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Find the probability that the toys are blue
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We
have one way or the other stated in light manner, the meaning of dependent events while
explaining the concepts of conditional event
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Event are said to be equally likely, when the occurence of one event is not more than the

occurence of other events in a given experiment
...
Therefore their probability will be respectively,
PROBABILITY AND SET THEORY
A set is a collection of object that are well defined and potrays an information in which they are
arranged
...
Probability can better be understood by expressing it in set language
represented by the (
)
...
Calculate for probability of the occurrence of 2
...
e
...
Associative law: the probability of events A,B and C are said to be associative when the
probability of A,B, and C occurring is the same or equal to the probability of A,C, and B
occurring
...
Commutative law: we are provided with two events A and B the probability of A or B
occurring is the same as the probability of B or A occurring
(
)
(
)
( )
( )
(
)
(
)
( )
( )

COMBINATION AND PERMUTATION
Combination and permutation is that aspect of probability calculation that deals with the
arrangement and selection of several set of events, items, or objects
...
Therefore, permutation and combination, explains, the number of ways an
operation can be performed with relation to multiplication theorem
...
These principles are:
The multiplication rule (r,s principle) : this states that if there are p ways of performing
one process or operation and q ways of performing the second operation
...

Example: if there are 3 different ways of kicking a ball, and 2 ways of throwing the ball,
the total number of ways of playing that ball is

Therefore,
= 3 2= 6
There are 6 different ways of playing the ball
...
It is simply calculated by
multiplying the newly introduced operation C to the product of those operations that have
occurred (AB)
FACTORIAL
Factorial n, written as
√ is defined mathematically as
)
If we are asked to calculate 5! It will be

(

)(

It is also used to express the number of ways a given set can be arranged
...
g: Show the possible ways we can arrange four different books in a shelf of four
different rows
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What if a given set is
arranged in r ways? The permutation will be represented by
(

)

Example: How many ways can we fill a vacant spot of 5 columns with 10 different
books?
n

Pr

(

)

10

P5

(

)

The above expression, explains the fact that the number of permutation or arrangement
(n) takes r at a time supposing the sets are unlike (not the same)
...
g
...

The permutation can be calculated by first calculating for 3 which occurred twice
(

)

( )

Therefore,

But in cases where repetitions is allowed, for instance if we are asked to find the arrangement of
2 things taken out of 6 unlike things when the repetition is allowed, it will be 6n =62 = 36 ways

PERMUTATION OF IDENTICAL OBJECTS
Supposing we are given a set of objects, and among that group, there exist objects which are
identical i
...
objects that look alike
...

R = number of identical object
...

Example 1
Supposing we are given the word;

RECONCILE
We are asked to permutate for it
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For instance, we are given;
2, 4, 7, 8, 2, 3;
We have 6 set of digits and 2 identical digits
...
The number of selecting n
unlike things taking r at a time is
(

)

(

(

) (

)

Example
Calculate for 8C4
(

(

)

)

Story 1:
After a president of a certain country won the election and was declared the now president of
the federation, he was formally sworn in few months later
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What they did was, to conspire with
the governors of the states of which their party was ruling
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The
plans they strategized failed woefully because they occupied the minor states while the
president’s party occupied the major states
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This time around, their plans were yielding
success because, the taut performed more than expected and were handsomely paid for their
impunity
...
He went ahead to seek advices from his special advisers
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While from 15 left, a committee of 4,
should be selected that will work side by side with security agencies in curbing the menace
...
Now, the question is this:

i)
ii)
iii)

How many ways can the president select a committee in charge of investigatory
cases?
How many ways can the president select a committee for security cases?
How many ways can the president select a committee for supervisory cases?

Solution

i)

The number of ways will be given as:

n= total number of persons available = 20
r =number of person selected for investigatory cases =5
20

C5

(


...

Example 1
How many numbers greater than 4,000 can be formed using some or all the digits 6,5,4,3,2
...
This means that there are 3 ways to
arrange 4 digits number which will be given as 4P3
Therefore, the total arrangement will be
3 4P3
But for the above problem to be solved conductively we will seek arrangements in which all
five digits are used and also one in which 4digits are used as well
...

Conditional permutation is all about placing restriction on the arrangement of object
...
That means, if restrictions are placed, the permutation will not be the same
as when no restrictions are placed
...
What do you suggest we do?
Let’s arrange the first L
...


P I A R
The arrow above indicates where the first L can possibly be fixed
...
What about the second L? Let us
assume we fixed the first L before P in the letters above,

L P I AR
Therefore, there are 4 possible positions/spaces left for the second L to be fixed
...
e
...

The permutation for those Ls is
But, we are asked to find the permutation for the set of letters “PILLAR” with the Ls not
coming together
...


Solution
Here, there are two restrictions placed;
i)
ii)

No number should be repeated
Numbers that are greater than 3,000 should be formed
...

2, 4, 5, 6, 7
`
Let us start with selecting numbers and arranging it to the left hand side
...
So, we are left with four chances,
4

Therefore, we are left with (2, 5, 6, 7), because, 2 can be fixed in any other position, other
than the first position from the left
...

4

4

3

2

Therefore, the total number of arrangement of digits greater than 3000, if no repetition is
allowed will be,

But permutation for those set of digits will be:



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