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STATISTICAL PROBABILITY
Objectives:
i)

To use statistics as a method in addressing problems related to
probability

ii) To understand the concept of normal distribution and how it can be explained
with a diagram
...

Introduction

Statistics can be seen as the collection, organization, analysis and interpretation of
numerical data
...
This data
collected can be presented on graph or charts to give the viewer an oversight view
of what the distribution looks like
...
While doing
interpretation, the reasons while that statistics was carried out will be sorted out
...
One of the vital processes is taking
the probability of circumstances surrounding data collection which may likely
occur
...

PROBABILITY DISTRIBUTION
Although, we have several probability distributions in statistics, such as;
i)
ii)
iii)
iv)
v)
vi)

Binomial distribution
Poisson distribution
Normal distribution
Multinomial distribution
Negative binomial distribution
Hyper geometric distribution
...


BINOMIAL DISTRIBUTION
The binomial distribution also known as “Bernoulli distribution
Is associated with a Swiss mathematician James Bernoulli also known
as Jacque (1654-1705)
(S
...

In each trial, there are only two possible outcomes, which are success(P)
and failure(q)’
2

iii)
iv)

The sum of the outcome is equal to one, (p + q = 1)
The outcomes of the trials are statistically independent, i
...
the outcome,
be it success/failure, do not affect the outcomes of the subsequent trials
...
e
...

It is mathematically expressed as:
( )

n

Cx

Where:
( )

Example 1 : A fisher man was able to catch 10 fishes, it was later found out
that, 40% 0f the fishes caught by him, were caught alive, then what is the
probability that:
i) More than 4 fishes will be caught alive
ii) Less than 4 fishes will be caught alive
iii) Exactly 5 fishes will be caught alive,

solution
i) for more than 4 will be:
( )
( )

( )

( ( )

( )

n

Cx

Where:
3

( )

( )

( ))

( )

10

( )

C0

Therefore: ( )
( )

n

( )

( )

( )

Cx

10

C1

( )
( )

10

C2

( )

10

C3

( )

10

C4

Therefore:
(

)

(
)

(

)

Therefore, the probability of more than four fishes to be caught alive is,
0
...
3824
...
It was developed by a French mathematician,
Simeon Denis Poisson, (1781-1840)
...

(S
...
It also describes the time concept of an event
...

Solution
5

i) the probability of less than 3 errors to be committed will be:
(

)

( )

( )

( )

( )
:
( )
( )
( )
( )
( )

( )
Therefore: (

)

ii) probability of more than 3 errors to be committed will be:
(
)
( )
( )
( ))
( ( )
Where:
( )
( )
Therefore:
6

(

)
(

(( ( )
)

( )

( ))

(

(

( ))
)

)

iii) for no error to be committed will be given as
( )

Probability cannot be conveniently taken if the statistical data is not normally
distributed
...
Let us closely discuss the above
mentioned concepts
...
It state how a distribution is supposed to look like under normal
circumstances
...
In a normal distribution most values (data) falls around the
mean, while those that fall far from the mean, either greater than the mean, or
lower than the mean are regarded as extreme values
...
The normal
distribution curve was drafted from the idea of frequency polygon
...

The area under the normal distribution curve can be given as:
√

Where:

(

)2

x = data distribution

= variance
= standard deviation
...
71828
...


8

20 32

But due to the complexity of normal distribution values, representing it on the
curve is an uphill task
...


Standard distribution: A distribution is said to be a standard distribution if the
standard deviation of the distribution is equal to one(1) and the mean of that
distribution is equal to zero(0)
...
Area under the standard curve is
equated to a unity(1) while the mean is rounded up to zero(0)
...
Converting
a normal distribution to a standard distribution can be done using the expression
below
...
25

Note that:
- After calculating the standard distribution(z), if negative(-) the value will fall
at the left hand side, but if positive(+) the value will fall at the right hand
side,
- While looking for the table value of the corresponding value of z, the
sign(+/-) before the value is ignored
...

- The corresponding table value must not be greater than one, must be
decimal value beginning with (0
...
E
...
3442 is
taken as 0
...
Lets draw back to the number line in mathematics,
because the idea of number line is important in solving similar questions
...

i
...
Before we proceed let us note the
following
(

(

)

(

)

(

)

)

The following are the rules of Probability of a standard curve are
Rules 1: to calculate the probability of a distribution being less than Z (P(x < -z),
first take the table value of z, and subtract the corresponding value from 0
...
5) where tz is table value of z
...
5
(

)

z
...
This is calculated by taken the difference
between their respective table values
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Rule 3: when we are ask to calculate the probability of the distribution that falls on
both right hand and left hand sides
...

(

)

13

-z 0 z
(

)

-z + tz

Rule 4: when we are ask to calculate for P( -z > x > + z), we will apply the first
(1st) rule and sum their results
...
5, calculate for *

14

)

0
0
...
5

0
...
5

ii) (

)

0 0
...
5

(

)

)

0

z

15

-0
...


0 1
...
5
*

2)

1
1=

1

2

0
...
4938
(

1

2)

Rule 3
Exampe: given that, z = -1
...
5
...
0 0
(

)

1

1
...
5, calculate for (

Example: given that z1 =

)

-1
...
5
)
)

LOGICAL PROBLEMS: Supposing a problem that is similar to real life
situation, synonymous to the distribution is encounted the best approach is to
convert the distribution to standard distribution using the expression

Then multiply the final result after looking for the corresponding table values, with
the total population of the distribution

17

Example: 20,000 students sat for General Examination
...
5 0
The probability of those passing the exam at below 15 years is
(

)

tz = 0
...

ii) Above twenty two years
...
08

(

)

z=

Therefore 9,362 students above the age of 22 are likely to pass the examination
...


-0
...

19

iv) Greater than twenty but less than twenty four
1

2

1

Z2

-0
...
25

2

2,612 students are expected to pass, at the age greater than 20 but less than 24
...

1

2

Z1
Z2

0 0
...
33

20

(

)

0
...
17

T0
...
1293
T0
...
0675
(

)

1,236 students that falls under that category are expected to pass the examination
...

1

2

-0
...
33

0
...
08

(

)

(

)

Therefore the total number of those that are likely to pass below the age of twenty
and above the age of twenty five will be 16,776 students
...
Many
businessmen, while expecting further success, makes decision that will aid them to
achieve their expectation
...
it is of great important to statistical work, because it enable
statistician to decide on the right step to take
...

The X is used to expressed discrete random variables which can be assume the
values of x1 x2 x3……………
...

If P1 + P2 + P3 + P4 + ………
...
PnXm

( )will be given as

( )

P1X1 + P2X2

The above expression show that the expected value ( ) will be equal to the sum
of each particular random value X, multiply by their respective probabilities that X
equate at that given time
...

Business;
A: has profit of $5,000 with a probability of 0
...
4
...
3, or a loss of $6,000 with a
probability of probability of 0
...

C: has a profits of $4,800 having a probability of 0
...
2
...

Solution
Mathematical expectation is given as
( )

P1X1 + P2X2 + P3X3………
...
Since the
maximum gain is $3060 which can only be achieve with business C, therefore, the
businessman is advice to pick-up business C, for maximum profit
...
He invested 20% of his pension on his
business, unfortunately for John, he failed and his business collapsed
...
John took a decision to
set-up another business, but this time, he decided that he would learn the business
from a close friend who was already into the business, in other to avoid a repeat of
circumstance
...
Armed with the necessary experience for
the new business, John decided to invest 35% of his pension which was 15%
greater than the rate invested earlier
...
Ironically, the business failed, for a
reason best known to his fate
...
John this
time thought to himself that he would not only receive ideas from his friend alone,
but he will also gain more ideas from those who have experiences on the business
of his dream
...
These proposals, their
projected profits and their respective probabilities were as stated below
...
23
that it will succeed
...
82
that it will succeed
...
44
that it will succeed
...
89
that it will succeed
...
23

X = 0
...
89

From the analysis above, it will be advisable for John to pick up business B,
because it has the highest profit
...

At this topic, we will discuss how certain statement made can be proven either true
25

or false, which will guide us in our decision making
...
This will lead us to what is called
a hypothesis
...
It is a tentative statement made by a scientist or a
researcher which is yet to be proven
...

Statisticians just like scientists make hypothesis about data distribution, which is
tested under certain specification with the knowledge of probability
...
Statisticians believe that the estimated value must
not be equal to the actual value, but it must fall at a range close to the actual level
for it to be accepted
...


SIGNIFICANT LEVEL
Significant level is that point at which an experiment either rejects or accepts a
hypothesis
...
The significant level
(normally expressed in percentage), determines if the hypothesis should be
accepted or rejected
...
Significant
level is complementing the confidence level
...
If it has 99% of
accepting a hypothesis, then it will have 1% of rejecting the hypothesis
...
This can be better explained in the
diagram below
...
05/2

region

0
...
If 100% is
rounded up to 1 then 5% will rounded up to 0
...
But since it is a two tailed curve
it will be divided by two and distributed at both tail regions
...
5 0
...
475, which occupies the acceptance region
showing the confidence level, so let us confirm what value in the table of area
under the curve has 0
...

0
...
96

Therefore the corresponding value of 0
...
96 while that of 0
...
58 if we follow the same process as was done in 5%
...
96, if not our hypothesis will be rejected
...
58, if not the hypothesis should be

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rejected
...


TYPES OF HYPOTHESIS
The major types of hypothesis are:
(i) The null hypothesis
(ii) Alternate hypothesis
The null hypothesis:- States that every hypothesis is null, equal, void until it
is proven otherwise
...
It is mathematically explained as
...

Alternate hypothesis, explains that there are differences between the estimated
value and the calculated value, until it is proven otherwise
...
There are certain errors committed while testing hypothesis
...

Type 2 error: - The hypothesis is false but our test accepts it
...
The
hypothesis to be rejected is the null hypothesis
...
These situations
involve:Test for the number of success
Test for the proportion of success
Test for difference in proportion
Test for mean
Before we proceed to the above stated concepts, we should bear in mind that in
most cases, samples are drawn from a population usually large, which can be used
for statistical experimentation as related to hypothesis testing and the final result is
used to judge the population as a whole
...
The null hypothesis at this level will hold that the probability of
success (p) must be equal to the probability of failure (q)

While the alternate hypothesis contradicts the null hypothesis with the view that
the probability of success will not be equal to the probability of failure
...
But if the difference between the probability of success and failure
respectively is greater than or equal to (i
...
Diff 1
...

The difference can be mathematically expressed as

Where S
...

q = probability of failure
p+q=l
q=l–p
Example 1:- A corn is tossed 200 times, 120 was obtained as heads
...
E = √npq
P = expected probability of head is ½

30

√

√

√

(i
...
it is expected that 100 heads will turn up for the corn to be unbiased)
`
Since the difference is greater than 1
...

Diagrammatized as

0

1
...
83

Step 3
The corn is said to be biased since the probability of success is significantly
different from the probability of failure
TEST FOR THE PROPORTION OF SUCCESS

31

We applied this method of testing hypothesis, when we decide to record the
proportion of success instead of the number of success i
...


of the number of

success in each sample
...
E

Where it is expressed in interval as
P

1
...
E)

P 1
...
E)

(5%)
(1%)

Example1 A trader claims that 21% of the tomatoes he bought where bad, when he
found out that 120 tomatoes out of 1,670 tomatoes were bad
...
88
Given the mathematical expression
q
Where
=√

S
...
96 (S
...
12

21% or 12% ????

=0
...
88

1
...
0079)

0
...
015
0
...
015
Estimate will fall at the range of;

0
...
895

The proportion of good tomatoes will be ranging from 0
...
895 while the
number of good tomatoes will be ranging from
0
...
895 (1,670)
1,445 to 1,495 and that of bad tomatoes will be
(1,670 – 1,495) to (1,670 – 1,445)
175 to 225
But the trader estimated 120 bad tomatoes as 12% meanwhile, 175 – 225
represent 12%
...


Example 2: A trader in oranges has long been in the business of selling oranges
for several years
...
They were so close that, the trader
sometimes buys oranges on credit and pays back any other time he comes for
purchases
...
What the trader did
was to divert to another customer from a different market altogether
...
Since he was new to his customer, he was compelled by circumstance to
pay for all the goods he bought without owing any
...
In
annoyance, the trader reaches the wholesaler which was his new customer through
phone call and complained that, 40% of the goods he bought were bad
...
Instead of the wholesaler to apologize, he went
ahead to argue with the trader, saying that he was wrong with his estimate
...


Therefore after due analysis, it is found out that the oranges that were bad falls
between 37
...
973% of the samples
...
The above analysis has proven that
the trader was right with his claim
...
We may use this
to know if the differences between their attributes are significant or not
...
E =√

(

)

X1 = number of occurrence in sample 1
X2 = number of occurrence in sample 2
N1= size of sample 1
N2 = size of sample 2
P=combined proportion of their attributes to the total

population
...
E
...
Can we say that at 5% significant level there is a significant
difference in both schools A and B as it concerns the proportion of reading habit?
Step 1
35

H0

There is no significant difference

H1

There is significant difference

Step 2

Given

√

(

)

X1 = 550
X2 = 980
n1 = 1000
n2 = 1300

√

(

)

(

√

√

)
(

√
36

)

Step 3:- Since it is greater than 1
...
This
further shows that Ho(null hypothesis) will be rejected, while the H1 (Alternate)
hypothesis will be accepted
...
This hypothesis
testing is of great importance to statistical work, and there are two basic tests for
mean
...
At this point, we should bear in mind that, we will
carry out test for the hypothesis of large samples
...
It was later found out that the mean score was 47%
...
25
38

Step lll: Since 1
...
58 (1% significant level), therefore there is no
significant difference between the two means
...

This is computed by taking the “t” table of (n – l), where n is the size of the sample
involved
...
If:
v

H0
H1

Tv
rejected
accepted

Cv Tv
H0

accepted

H1

rejected

Where
Cv
Tv

table

But while looking for the table value, the percentage of the significant level is
rounded up to a unity (1) which is (
)
...

Example: - The mean score of a given distribution of 20 is estimated as 14% at
95% confident level
...
Can we say that, there is a
significant difference between the two means, given a standard deviation of 8%
and the sum of the squared (solution????) standard deviation 6
...
23
X = estimated mean
M = actual mean
N = sample size

√

Calculated value = 2
...
05 19 = 2
...


Example 2:- A producer of phone batteries claims that his batteries have a mean
life of 20hours when fully charged, with a standard deviation of 8hours
...

Life in hours …
...


Step II
Calculated value

√

=√

√

√

Calculated value
Table value (n

)

t0
...
182

Step III
H0

accepted since the calculated value is less than the table value
H1

rejected
...
7)
P(x > 1
...
7 < x < +2
...


2) An amount of money is given to villagers residing in a particular village
...

What will be the amount given to those villagers that falls (under) in these
groups,
i) Above 20 years
ii) Below 28 years
iii) Above 26 years,
iv) Between 27 and 29 years,
v) Between 24 and 26 years, will be paid
...
02, and a profit of $2million dollars
...

5) Town A has 2,500 youths, with 700 youths as smokers, town B has 2,900
youths with 1,100 youths as smokers, can we say there is a significant
difference in the smoking habits of youths of both towns, as the proportion of
smoking is concerned?( Assume 5% significant level)
...
9554
2) 0
...
9447
4) I)$13,250,000 ii)$9,740,000
iii)$8,333,333
iv)$1,276,000
iv) $40,000
v)$1,276,000
...
GODMAN{2008) “permutation and combination”longman publishers, Nigeria
...

S
...

S
...

S
...
K Gupta, sultan chand &
son publishers, India

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