STPM MATHEMATICS (T) ASSIGNMENT C 2012-2013 Statistical Inferences on the Distribution of Digit in Random Numbers by Stephen, P. Y. Bong

MALAYSIA HIGHER SCHOOL CERTIFICATE (STPM)

954/4 MATHEMATICS (T) (PAPER 4)

THIRD TERM: STATISTICS

ASSIGNMENT C: MATHEMATICAL INVESTIGATION

(2012/2013)

STATISTICAL INFERENCES ON THE DISTRIBUTION OF DIGIT IN

RANDOM NUMBERS

By

Stephen, P. Y. Bong

(September 2013)

MALAYSIA EXAMINATION COUNCIL

Statistical Inferences on the Distribution of Digit in Random Numbers

by Stephen, P. Y. Bong (September 2013)

i

DECLARATION

I hereby declare that this report entitled “Statistical Inferences on the Distribution of Digit

in Random Numbers” is the result of my own work except for quotations and citations

which have been duly acknowledged.

Name : Stephen, P. Y. Bong

Email : [email protected]

Date : 21 September 2013



ii

TABLE OF CONTENTS

DECLARATION ....................................................................................................................... i

LIST OF FIGURES ................................................................................................................ iii

LIST OF TABLES .................................................................................................................. iii

ABSTRACT ............................................................................................................................. iv

1.0 INTRODUCTION ........................................................................................................ 1

1.1 Aims and Objectives ............................................................................................. 2

1.2 Outline of Report .................................................................................................. 3

2.0 METHODOLOGY ....................................................................................................... 4

2.1 The Generation of Thirty 3-Digit Numbers by Drawing of Poker Cards from

French Deck .......................................................................................................... 4

2.2 The Generation of One Hundred 3-Digit Numbers by Computer Algebra System

(CAS) – Wolfram Mathematica 7.0 ...................................................................... 5

2.3 The Revise and Comparison of Probabilities Obtained by Simple Experiment

(Drawing of Cards from French Deck) by Interval Estimation ............................ 8

2.4 The Sampling of Fifty 3-Digit Random Numbers by Random Number Tables ... 8

2.5 The Sampling of Sixty Four 3-Digit Numbers Generated by the Function

51000 n .............................................................................................................. 9

3.0 RESULT ANALYSIS AND DISCUSSIONS ........................................................... 10

3.1 The Analysis of Thirty 3-Digit Numbers Obtained from Simple Experiment

(Drawing of Cards from French Deck) ............................................................... 10

3.2 The Analysis of One Hundred 3-Digit Numbers that are Randomly Generated by

Wolfram Mathematica 7.0 .................................................................................. 11

3.3 The Analysis of Fifty 3-Digit Numbers by Random Number Tables ................. 13

3.4 The Analysis of Sixty Four 3-Digit Random Numbers Generated by the Function

51000 n ............................................................................................................ 14

4.0 CONCLUSIONS ......................................................................................................... 17

5.0 REFERENCES ........................................................................................................... 18

APPENDICES ....................................................................................................................... A1

Appendix 1 – Computational Codes............................................................................ A1

Appendix 2 – List of Statistical Tables ....................................................................... A2

Appendix 3 – Screenshots of Random Numbers Generated from Wolfram

Mathematica 7.0 ............................................................................................... A11



iii

LIST OF FIGURES

Figure 1: French Deck used in the simple experiment to generate thirty 3-digit numbers ........ 5

Figure 2: The computer algebra system (CAS) - Wolfram Mathematica 7.0 used in this work

to generate random numbers .......................................................................................... 6

Figure 3: An illustration of the selection of random numbers from a portion of Random

Number Tables ............................................................................................................... 9

Figure 4: The screenshot of random generation of one hundred 3-digit numbers ranged from 0

to 999 by Wolfram Mathematica 7.0 ........................................................................... 11

Figure 5: The screenshot of random generation of sixty four real numbers ranged from 0 to 1

by Wolfram Mathematica 7.0 ....................................................................................... 11

LIST OF TABLES

Table 1: Confidence intervals for the population proportions, p (Crawshaw & Chambers 2002)

.................................................................................................................................................... 7

Table 2: Outcomes obtained from the drawing of cards from French Deck ............................ 10

Table 3: Frequencies and probabilities of 3 different digits, 2 same digits, and 3 similar digits

.................................................................................................................................................. 10

Table 4: The frequencies and proportions computed based on the sample of one hundred 3-

digit random numbers ................................................................................................... 11

Table 5: The symmetric 90% and 95% confidence intervals for the probabilities that a 3-digit

number has three different digits, two same digits and three identical digits .............. 12

Table 6: Frequencies and probabilities correspond to each respective category in a sample of

fifty 3-digit random numbers ....................................................................................... 13

Table 7: The observed frequency and the expected frequency for Chi-square test ................. 14

Table 8: Processing of raw numbers into a sample of sixty four 3-digit random numbers ..... 15

Table 9: Random Number Tables ........................................................................................... A2

Table 10: Percentage Points of the 2 Distribution ............................................................... A7

Table 11: The Upper Tail Probabilities z of the Standard Normal Distribution X~N(0, 1)

................................................................................................................................................. A9



iv

ABSTRACT

The information perceived and processed by human‟s cognitive system may be simple

or complex, clear or distorted, and complete or filled with gaps. Yet the options available are

often subjected to uncertainties and natural variable which are unpredictable, thus subjective

probability has been extensively employed in decision making. The outcomes and results are

often validated by the aid of inferential statistics in which inferences and conclusion can be

drawn based on the estimation of population parameters and the test of hypotheses.

Consequently, mathematical investigations on the distribution of digit in one thousand 3-digit

numbers ranged from 0 to 999 are conducted. Due to the immense size in terms of population,

it would be a tedious task if census on each data is conducted. Hence, samples with sizes of

30, 100, 50 and 64 are obtained through various methods such as simple experiment, number

generator and random sampling. The results obtained from the analyses conducted on each of

the sample have a high resemblance in terms of probability in which the discrepancy is not

exceeded by 0.15. In addition, the testing of hypotheses drawn by one-tail test and Chi-square

goodness of fit test also concluded that the suggested probability in which a 3-digit number

encompasses of 3 distinct digits is 0.73 and 0.7 are accepted with clear justification. As a

result, in order to getting a sample which possesses the capability to provide a better

indication to population parameters with higher accuracy, a sample with larger size which has

a range 100 < n < 400 is recommended.



INTRODUCTION 1

1.0 INTRODUCTION

The information perceived and processed by human‟s cognitive system may be simple

or complex, clear or distorted, and complete or filled with gaps (Chai 2012). Hence,

Kahneman and Tversky (1972) addressed that subjective probability has been extensively

employed in decision making. This is due to the options available are annexed with

uncertainties and variations. As a result, statistical inferences such as estimation and

hypothesis test (Soon & Lau 2013) could be conducted in the examination of corresponding

attributes and outcomes associated (Wallsten & Diederich 2011). As mentioned by Machina

and Schemeidler (1992), although there are several literatures that done beforehand by

theorists and philosophers such as Koopman (1940) and Ramsey (1931) in which definitions

of probability are proposed, and yet the shortage of unique phrase and concise definitions in

which subjective probability can be explicitly described in detail still exists. In a nutshell,

according to Anscombe and Aumann (1963), the term subjective probability can be

interpreted as “a person‟s preferences, in so far as these preferences satisfy certain

consistency assumptions”. In terms of simplicity, it could be defined as the likelihood that a

particular outcome will occur based on individual judgment or degree of belief (Clemen 1997).

Besides, according to Kyburg (1978), the computations of subjective probability of an event

do not possess a standard mathematical equation or formula as it encompasses higher

frequency of bias.

Apart from branches of psychological science such as cognition as well as decision

making, the advances and rapid development of mathematical statistics have also contributed

subjective probability to be extensively utilized in real-life and industrial applications.

Biotechnological industry and forensic science are typical pervasive fields in which subjective

probability is extensively intervened. For instances, the research conducted by Biedermann

and colleagues (2013) reveals that law and its interface with forensic science represent an

illustrative example of an area of application where decision-making plays a central role.

Probabilistic inference is a part of this framework, but only as a preliminary step which based

on one‟s beliefs. Subsequently, a decision is made and a conclusion or verdict is drawn.

Besides, Velasco (2012) and Weir (2013) also clearly illustrate that in their researches,

subjective probability is widely applicable in the processing of Deoxyribonucleic acid (DNA)

and gene expression. This is due to the fact that gene expression would be inderministic if the

question of whether a gene will be expressed in a given cell in a given time frame is not

determined even by the exact state of all the cellular and environmental components at a given

time. In addition, some widespread situations in real life in which subjective probability is

involved are presented in the subsequent paragraph.

The probability that double Ace could be drawn from French Deck in a Game of

Blackjack (or often referred as The Twenty-One) is 7.51% with a constraint of the dealer

stand on all 17s (Blackjack Info 2013; Maskalevich 2011). This is owing to the existence of

chances of drawing cards with odd ranks out of the four French suits could constitute to a sum

of twenty-one in the game (Christenstock 2013). In addition, the chances of an election

candidate from a political party could win in a general election and become a Member of

Parliament (MP) is 33%. This is due to the number of options in terms of political parties

involved in the general election (Jacob 2011). Apart from that, one‟s could also deduce the

probability in which one of the columns in a structure will experience buckling is 0.23%

(Hibbeler 2009), since parameters such as settlement of soil (Whitlow 2000), compression

property of distinct alloy steels (Benham et al. 1987), and the vibration induced by wind loads



INTRODUCTION 2

(Thomson & Dahleh 1998) will also deviate the occurrence of buckling drastically. Moreover,

one‟s could overthrow or oppose the hypothesis created by an experienced engineer that the

time taken for an infinitesimal sand to reach the ground level from the top of a construction

building is 10 minutes. This is due to the fact that the induced drag and viscosity of air at 1

atmosphere will result in variation of terminal velocity which might prolonged the settling

time (Wong 2012).

Based on the four real-life situations in which subjective probability is intervened as

mentioned above, it can be distinctly seen and concluded that experiences and personal‟s

degree of belief act as a dominant role in the determination of subjective probability.

Therefore, Crawshaw and Chambers (2002) addressed that techniques of statistical inferences

such as sampling and estimation as well as the test of hypotheses are frequently conducted to

validate the outcomes and results obtained. Consequently, in order to understand and

familiarize the fundamental mechanics and physics behind subjective probability, a

mathematical investigation on the distribution of digit in one thousand 3-digit numbers ranged

from 0 to 999 has constituted the primary intention of this work. This distribution of digit in

random numbers is taken into considerations in this work as it is in fact a strong of the 10

digits (0-9) arranged in an irregular way. As mentioned by Bafna and Kumar (2012), although

the term random in random number reflects the irregularities in terms of the arrangement of

digit, but it is always emphasized that random numbers cannot be truly random since they are

generated by a fixed algorithm or programming code such as the linear congruence method

for generating pseudo random numbers which is extensively applied in linear recurrence

relation (Marsaglia & Zaman 1993): xi+1 ≡ axi + b mod M, where a and b are arbitrary

constants, M is the element in the set of real numbers in the closed interval from 0 to 1, and

0,i . As a result, it would be an interesting intellectual and an enjoyable process in the

pathway of conducting analysis and mathematical investigation on the distribution of digit in

random numbers.

1.1 Aims and Objectives

As mentioned above, since the nature of subjective probability are permeated with

uncertainties, and it is a strong dependent variable corresponds to past experiences and

personal‟s degree of belief, therefore, it is an indispensable predecessor to possess a sound

comprehending on the fundamental mechanics on inferential statistics in which conclusions

and verdict could be drawn. Hence, the specific aims and the corresponding objectives of this

work are listed below according to the point of indentation:

To visualize the pattern of arrangement of digit in random numbers.

To investigate the distribution of digit based on sample statistics corresponds to

a population of one thousand 3-digit numbers ranged from 0 to 999.

To observe the frequency distribution corresponds to the samples of random

number obtained with classifications of 3 distinct digits, 3 identical digits, and 2

similar digits.

To conclude or make a statistical inference on the population of one thousand 3-digit

numbers based on statistics of samples.



INTRODUCTION 3

To determine the sample statistics which will accurately reflects the

characteristics of the entire population (one thousand 3-digit numbers) interval

estimation and hypothesis testing.

1.2 Outline of Report

The methodologies and approaches employed in the solving of problems in this work

are reviewed in Section 2.0. Whereas, Section 3.0 outlines the results and discussions on the

outcomes obtained. Finally, draw of conclusions for the entire assignment is presented in

Section 4.0.



METHODOLOGY 4

2.0 METHODOLOGY

As mentioned in Section 1.0, since past experiences and one‟s degree of belief play a

significant role and act as the determinism factors which would drastically varies the

outcomes and results of subjective probability, and in order to increase its degree of

credibility, the results obtained are often validated by the aid of methodologies such as

interval estimation and hypothesis testing. Such techniques from statistical inferences are

adopted are due to considerations of time and costs (Soon & Lau 2013). Apart from that, as

mentioned by Crawshaw and Chambers (2002), it would be a tedious survey if the size of

targeted population for census to be conducted is large. On the contrary, although numerous

advantages as mentioned could be resulted through the application of sampling and estimation,

but, biases such as uncertainties and natural variations could significantly affect the

characteristics of population parameters. Hence, in order to obtain a good sample in which the

entire population can be represented, the methods used to obtain the samples in this work, and

the approaches taken in the analyses of outcomes and results are presented in the subsequent

subsections.

2.1 The Generation of Thirty 3-Digit Numbers by Drawing of Poker Cards from

French Deck

Since a deck of poker card (or often referred as French Deck see Figure 1 below)

encompasses of 52 cards which are made up from 13 distinct ranks with four of each French

suits namely: Diamond (♦), Club (♣), Heart (♥), and Spade (♠). By using “10”(s) as 0,

“Ace”(s) as 1, “2”(s), “3”(s), “4”(s), “5”(s), “6”(s), “7”(s), “8”(s), and “9”(s) correspond to

integers ranged from 2 to 9 respectively, a 3-digit number is obtained by drawing of three

cards one-by-one, without replacement. If cards such as “J”(s), “Q”(s), “K”(s) and “Joker”(s)

are drawn, it will be neglected until those cards which had been mentioned in the preceding

sentence are obtained. In order to ensure all the cards are randomly distributed, shuffling of

cards is performed before each of the subsequent draw. The outcomes are then tabulated in

Table 2 (see Section 3.1). Based on the outcomes tabulated in Table 2, the frequencies of

numbers with 3 different digits, 2 similar digits, and 3 identical digits are counted. Lastly, the

probabilities of getting each of the categories mentioned are determined by Eq. (1), and a

deduction on the nature of probabilities obtained is made.

nP

n

X xX x

S

Eq. (1)

where X represents the number of identical digit (X = 0, 2 and 3)

S is the sample space, n(S) = 30.



METHODOLOGY 5

Figure 1: French Deck used in the simple experiment to generate thirty 3-digit numbers

2.2 The Generation of One Hundred 3-Digit Numbers by Computer Algebra System

(CAS) – Wolfram Mathematica 7.0

In order clearly illustrates the distribution of digit in random number, as well as to

obtain a result which can give a more accurate indication of the population characteristics

being studied, a sample of one hundred 3-digit numbers are generated randomly by the aid of

computer algebra system (CAS) – Wolfram Mathematica 7.0 as shown in Figure 2. The

computational code used to generate one hundred 3-digit random numbers can be found in

Appendix 1. Wolfram Mathematica 7.0 is adopted as the random number generator in this

work. This is due to the equitability of the software in Swinburne University of Technology

(Sarawak Campus). Besides, it is also a trending and powerful computer algebra system

which has been extensively used by researchers and statistician nowadays, in such a way that

the randomness of the random integers generated can be assured. The outcomes are then

tabulated in Table 4 (see Section 3.2) and the frequencies of each case (distribution of digit)

are counted by the aid of tally. With the frequencies of each case known, the proportions (or

relative frequency of occurrence) are determined. Since the sample size is 100 (n ≥ 30), thus

the sampling distribution can be approximated by a normal distribution by the employment of

Central Limit Theorem (Crawshaw & Chambers 2002) as shown in Eq. (2).

N ,s

pqP p

n

Eq. (2)



METHODOLOGY 6

where p is the proportion of successes in the population

q = 1 – p and n is the sample size

Figure 2: The computer algebra system (CAS) - Wolfram Mathematica 7.0 used in this work

to generate random numbers

Usually, a continuity correction factor of 12n

is added when normal approximation to

the binomial distribution is considered. On the contrary, when significance test is

implemented and confidence interval approach is used, it is perfectly reasonable to specify the

confidence coefficient in advance at some conventional values such as 90% and 95%. Hence,

the approximate limits using the continuity correction also tend to be conservative

(Mendenhall et al. 2013). With the continuity correction neglected, interval estimation is

conducted based on symmetric 90% and 95% confidence intervals in such a way that the

population parameters can be accurately indicated and reflected since it possesses the

capability of providing a range of values which has certain probability of containing the

population parameter (Elsevier Inc. 2012). Apart from that, the results from interval

estimation are used to revise the probability computed from the sample of thirty 3-digit

numbers obtained by drawing of poker cards. The standard equation for the determination of

confidence limits is provided in Eq. (3) as follows. Lastly, a comment on the population

proportion is made.

2 2

Lower Confidence Limit Upper Confidence Limit(LCL) (UCL)

1 1,

s s s s

s s

p p p pp z p z

n n

Eq. (3)



METHODOLOGY 7

where 2

z is the critical z-values in confidence intervals. Table 1 below summarized the confidence intervals for the population proportions, p.

Table 1: Confidence intervals for the population proportions, p (Crawshaw & Chambers 2002)

90% 95% 99%

Normal

Distribution

Curve

α 10% = 0.10 5% = 0.05 1% = 0.01

Critical

z-values

(see Appendix 2 for

complete table)

The upper tail probability is 0.05, so the

lower tail probability is 0.95.

0.10

2

0.05

1

0.05

P 0.95

0.95

0.95

1.645

Z z

z

z

The upper tail probability is 0.025,

so the lower tail probability is

0.975.

0.05

2

0.025

1

0.025

P 0.975

0.975

0.975

1.96

Z z

z

z

The upper tail probability is 0.005, so

the lower tail probability is 0.995.

0.01

2

0.005

1

0.005

P 0.995

0.995

0.995

2.576

Z z

z

z

Confidence

intervals


1.645 , 1.645s s s ss s

p q p qp p

n n


1.96 , 1.96s s s ss s

p q p qp p

n n


2.576 , 2.576s s s ss s

p q p qp p

n n

Width 2 1.645 s sp q

n 2 1.96 s sp q

n 2 2.576 s sp q

n



RESULT ANALYSIS AND DISCUSSIONS 8

2.3 The Revise and Comparison of Probabilities Obtained by Simple Experiment

(Drawing of Cards from French Deck) by Interval Estimation

In order to revise the compare the probabilities obtained by drawing of poker cards

from French Deck, the standard error of the sample with size of 30 is computed. In addition, a

comparison on the probability distribution is conducted based on the results obtained from

interval estimation as tabulated in Table 4. If the revised probabilities are found beyond or

outside the confidence limits, the experiment will be repeated until the sample statistics could

fit the symmetric 90% and 95% confidence intervals.

2.4 The Sampling of Fifty 3-Digit Random Numbers by Random Number Tables

A sample of fifty 3-digit random numbers is obtained by the method of simple random

sampling. This is done in such a way that the numbers are obtained from Random Number

Tables (see Appendix 2) since the randomness of the sample can be assured. By selecting the

5th

, 10th

, 15th

, 20th

, and 25th

rows, a sample of fifty 3-digit random numbers could be obtained.

On the other hand, the numbers fall outside the range will be neglected. For the sake of

sensible visualization, an example of the selection of random numbers is presented based on a

portion of Random Number Tables as depicted in Figure 3. Hence, according to Figure 3, the

random numbers selected from the 10th

Row are listed as follows:

28 55 53 09 48 86 28 30 02 35 71 30 32 06 47 10th

Row

Thus, the 10 random numbers after processed are:

{285, 553, 094, 886, 283, 002, 357, 130, 320, 647}

Once the sample of fifty 3-digit random numbers is obtained by the methodology

mentioned above, the data is then tabulated in Table 5 (see Section 3.3) and the frequency of

each category is determined. Besides, the proportion of each category is computed as well.

With both the frequency and proportion known, a table which encompasses of observed

frequency and expected frequency is constructed. This is done in such a way that Chi-square

test can be conducted to determine whether the sample of fifty 3-digit numbers fit the

distribution of the revised probability. Whereby, the test statistic or often termed as the

Pearson Chi-square statistic is given by Eq. (4), and with the following null and alternative

hypotheses:

H0: The distribution of digit obtained by the sample of fifty 3-digit numbers fit the

distribution of the revised probability.

H1: The distribution of digit obtained by the sample of fifty 3-digit numbers does

not meet the distribution of the revised probability.

2

2 i i

i

O E

E

Eq. (4)

where Oi is the observed frequency (The frequency computed directly from the sample data

by tally method)




Ei is the expected frequency, and it can be determined by multiplying the proportion

with the sample size, Ei = np and n = 50.

Since the proportions of the population, p is known, then the degree of freedom is

given by v = n – 1 (Crawshaw & Chambers 2002), where n = 3 due to three categories of data.

Thus, the Chi-square goodness of fit test is performed at 5% significance level with degree of

freedom of 2 which corresponds to a critical value of 5.991 (see Appendix 2). If 2 is less than

5.991, then there is sufficient evidence to reject the null hypothesis. On the contrary, if 2 is

found greater than 5.991, then it can be concluded that the distribution of digit obtained by the

sample of fifty 3-digit numbers fit the distribution of the revised probability.

Figure 3: An illustration of the selection of random numbers from a portion of Random

Number Tables

2.5 The Sampling of Sixty Four 3-Digit Numbers Generated by the Function 51000 n

A sample of sixty four numbers ranged from 0 to 1 is generated randomly by the aid of

Wolfram Mathematica 7.0. The irrational numbers generated are then substituted into the

function 51000 , : 0,1f n n n . The computations are carried out by the aid of

Microsoft Excel 2013, and only the integer part of those numbers is taken into considerations.

The computational codes are attached in Appendix 1. The probability of getting a 3-digit

number with three different digits will be different from the revised probability. However, the

deviation shall not exceed by 0.1; therefore, the suggested probability of obtaining a 3-digit

number with three distinct digits is 0.73. Consequently, the null and alternative hypotheses are

stated as follows:

H0: P(X = 0) = 0.73

H1: P(X = 0) > 0.73

Since the primary intention of the hypothesis test conducted here is to determine whether the

suggested probability will be accepted, thus, it is a one-tail test with α = 0.05 and α = 0.10 at

significance level of 5% and 10 % respectively.




3.0 RESULT ANALYSIS AND DISCUSSIONS

The computations and analysis of results in the following subsections are based on the

methodologies proposed in Section 2.0.

3.1 The Analysis of Thirty 3-Digit Numbers Obtained from Simple Experiment

(Drawing of Cards from French Deck)

The outcomes obtained from the drawing of poker cards from French Deck are

tabulated in Table 2 below.

Table 2: Outcomes obtained from the drawing of cards from French Deck

No. Drawing of Poker Cards 3-Digit

Numbers No.

Drawing of Poker Cards 3-Digit

Numbers 1st Draw 2

nd Draw 3

rd Draw 1

st Draw 2

nd Draw 3

rd Draw

1 4 5 5 455 16 7 4 2 742

2 0 5 7 57 17 9 0 4 904

3 6 6 6 666 18 5 4 2 542

4 6 8 8 688 19 8 4 8 848

5 3 5 5 355 20 7 7 5 775

6 4 1 2 412 21 1 8 5 185

7 8 8 5 885 22 9 0 0 900

8 3 2 0 320 23 1 6 4 164

9 7 2 5 725 24 4 8 5 485

10 3 8 9 389 25 9 3 8 938

11 6 1 4 614 26 6 4 5 645

12 6 8 5 685 27 1 5 5 155

13 7 0 1 701 28 6 2 3 623

14 9 1 8 918 29 7 5 6 756

15 7 3 8 738 30 0 3 8 38

Based on the outcomes tabulated in Table 2 above, the frequencies and probabilities for 3

different digits, 2 same digits and 3 similar digits are computed and tabulated in Table 3.

Table 3: Frequencies and probabilities of 3 different digits, 2 same digits, and 3 similar digits

X = x Outcomes corresponding to X = x Frequencies

f

Probabilities

P(X = x)

0

(6,1,4), (1,8,5), (0,5,7), (6,8,5), (7,0,1), (1,6,4), (9,1,8),

(4,8,5), (7,3,8), (9,3,8), (4,1,2), (7,4,2), (6,4,5), (9,0,4),

(3,2,0), (5,4,2), (6,2,3), (7,2,5), (7,5,6), (3,8,9), (0,3,8)

21 0.7

2 (4,5,5), (9,0,0), (6,8,8), (3,5,5), (8,8,5), (1,5,5), (8,4,8),

(7,7,5) 8 0.266666667

3 (6,6,6) 1 0.033333333




According to the frequencies and probabilities for each case tabulated in Table 3, it

can be deduced that the probability of getting a number with three identical digits is extremely

low (i.e. P(X = x) = 0.0333) as compared to the probabilities of the other two categories. On

the contrary, the results also reveals that obtain a number with three different digits is the

easiest.

3.2 The Analysis of One Hundred 3-Digit Numbers that are Randomly Generated by

Wolfram Mathematica 7.0

The samples of one hundred 3-digit numbers that are randomly generated by the aid of

Wolfram Mathematica 7.0 are listed below:

765 905 863 718 518 307 166

002 156 727 556 699 813 056

216 091 078 096 777 996 790

362 171 666 294 856 819 951

324 404 562 775 626 255 518

177 980 760 795 392 917 916

407 281 396 245 042 820 812

743 810 953 228 037 168 861

388 393 695 869 802 405 409

991 118 768 976 080 809 388

Based on the data listed above, the frequencies, f and proportions, ps for each case are counted

and tabulated in Table 4 as follows:

Table 4: The frequencies and proportions computed based on the sample of one hundred 3-

digit random numbers

X = x Frequencies, f Proportions, sp

0 69 0.69

2 29 0.29

3 2 0.02

With both the frequencies and proportions known, the symmetric 90% and 95% confidence

intervals for the probabilities that a 3-digit number has three different digits, two same digits

and three identical digits are determined and tabulated in Table 5.




Table 5: The symmetric 90% and 95% confidence intervals for the probabilities that a 3-digit number has three different digits, two same digits

and three identical digits

Symmetric 90% Confidence Interval

Category 3 Different Digits (X = 0) 2 Same Digits (X = 2) 3 Same Digits (X = 3)

Proportion, ps 0.69 0.29 0.02

Confidence Interval

0.69 0.310.69 1.645

100

0.6139, 0.7661

0.29 0.710.29 1.645

100

0.2154, 0.3646

0.02 0.980.02 1.645

100

0.003, 0.04303

Symmetric 95% Confidence Interval

Category 3 Different Digits (X = 0) 2 Same Digits (X = 2) 3 Same Digits (X = 3)

Proportion, ps 0.69 0.29 0.02

Confidence Interval

0.69 0.310.69 1.96

100

0.5994, 0.7806

0.29 0.710.29 1.96

100

0.2011, 0.3789

0.02 0.980.02 1.96

100

0.074, 0.0474




The results tabulated in Table 3 and

Table 5 clearly manifested that there is a high resemblance in terms of proportions

obtained between two samples of distinct sizes. For instances, the deviation in terms of

probabilities from two samples that a 3-digit numbers has three distinct digits is only 0.01,

which is relatively small. Hence, it can be said that a larger sample would results in a more

accurate indication on the population parameter (Soon & Lau 2013). Apart from that,

although the probability that a 3-digit number has three identical digits is 0.02, but the results

obtained from the interval estimation distinctly illustrate that the lower confidence limit for

both symmetric 90% and 95% confidence intervals are -0.003 and -0.0074 respectively, in

which the probabilities are negative that is impossible to occur in real-life situations. In

addition, the occurrence of probabilities with negative values also implies that although the

true population parameters will be included if an interval with larger size is employed, but it

might leads to discrepancies of results as well (Crawshaw & Chambers 2002). Therefore, in

order to overcome the drawbacks of interval estimation, a larger sample with size of 100 < n <

400 would be recommended.

3.3 The Analysis of Fifty 3-Digit Numbers by Random Number Tables

Based on the tabulation of results for both symmetric 90% and 95% confidence

intervals in Table 5, it could be clearly seen that none of the probability in Table 3 lies outside

the confidence limits. Thus, the subjective probability obtained by drawing of poker cards

from French Deck in Section 3.1 is acceptable. Consequently, this constituted the formulation

of the following null and alternative hypotheses.

H0: The distribution of digit obtained by the sample of fifty 3-digit numbers fit the

distribution of the revised probability.

H1: The distribution of digit obtained by the sample of fifty 3-digit numbers does

not meet the distribution of the revised probability.

According to the approaches proposed in Section 2.4, fifty 3-digit random numbers are

selected from Random Number Tables and listed below:

555 956 356 438 548 246 223 162 430 990

576 086 324 409 472 796 544 917 460 962

378 594 351 283 395 008 304 234 79 688

311 693 324 350 278 987 192 015 370 049

299 498 942 468 496 910 825 375 919 330

Hence, the frequencies and probabilities correspond to the respective categories are computed

and tabulated in Table 6 below.

Table 6: Frequencies and probabilities correspond to each respective category in a sample of

fifty 3-digit random numbers

X = x Frequencies, f Probabilities, P(X = x)

0 40 0.8

2 9 0.18

3 1 0.02




With the observed frequency in Table 6 known, the expected frequency are calculated and

tabulated in Table 7.

Table 7: The observed frequency and the expected frequency for Chi-square test

X = xi

Observed

Frequency

Oi

Revised

Probability

p

Expected

Frequency

Ei = np i iO E

2

i i

i

O E

E

0 40 0.7 35 5 0.7143

2 9 0.2667 13.335 -4.335 1.4092

3 1 0.0333 1.665 -0.665 0.2656

50iO 50iE

2

2.3891i i

i

O E

E

As mentioned in Section 2.4, since the upper point critical value for Chi-square goodness of

fit test at 5% significance level with 2 degree of freedom is 2

0.05,2 5.991 (see Appendix 2),

which is much more greater than the test value of 2 2.3891 , therefore, there is no

sufficient evidence to reject the null hypothesis. As a verdict, it can be concluded that the

distribution of digit obtained by the sample of fifty 3-digit numbers fit the distribution of the

revised probability.

3.4 The Analysis of Sixty Four 3-Digit Random Numbers Generated by the Function51000 n

The sixty four real numbers ranged from 0 to 1 that are randomly generated by Wolfram

Mathematica 7.0 are listed below:

0.708477 0.295863 0.971974 0.099977 0.962334 0.630794 0.379602 0.900808

0.217191 0.914257 0.156891 0.405242 0.23194 0.252731 0.369021 0.630475

0.845501 0.307553 0.397659 0.211588 0.993561 0.394024 0.297604 0.17321

0.428942 0.179826 0.291179 0.271801 0.250618 0.022988 0.760863 0.013218

0.904237 0.497101 0.001202 0.869522 0.927179 0.410847 0.135921 0.837393

0.760437 0.253564 0.639631 0.981092 0.06209 0.578899 0.994646 0.773488

0.669724 0.980035 0.180364 0.008676 0.898413 0.776303 0.951459 0.435168

0.367291 0.138426 0.519327 0.411859 0.848798 0.819825 0.566698 0.132321




By the aid of Microsoft Excel 2013, the raw numbers listed above are substituted into the

function 51000f n n , computed and tabulated in Table 8.

Table 8: Processing of raw numbers into a sample of sixty four 3-digit random numbers

No.

Raw

Random

Number

n

f(n) int(f(n)) No.

Raw

Random

Number

n

f(n) int(f(n))

1 0.708477 933.3943 933 33 0.962334 992.3507 992

2 0.217191 736.8318 736 34 0.23194 746.5779 746

3 0.845501 966.9919 966 35 0.993561 998.7089 998

4 0.428942 844.2668 844 36 0.250618 758.2326 758

5 0.904237 980.0686 980 37 0.927179 984.992 984

6 0.760437 946.7006 946 38 0.0620896 573.5929 573

7 0.669724 922.9521 922 39 0.898413 978.8028 978

8 0.367291 818.4687 818 40 0.848798 967.7449 967

9 0.295863 783.8232 783 41 0.630794 911.9636 911

10 0.914257 982.2311 982 42 0.252731 759.5069 759

11 0.307553 789.9216 789 43 0.394024 830.0505 830

12 0.179826 709.5296 709 44 0.0229878 470.2183 470

13 0.497101 869.5387 869 45 0.410847 837.0204 837

14 0.253564 760.0069 760 46 0.578899 896.4386 896

15 0.980035 995.9747 995 47 0.776303 950.6185 950

16 0.138426 673.3541 673 48 0.819825 961.0461 961

17 0.971974 994.3309 994 49 0.379602 823.8833 823

18 0.156891 690.4299 690 50 0.369021 819.2382 819

19 0.397659 831.5764 831 51 0.297604 784.7435 784

20 0.291179 781.3255 781 52 0.760863 946.8066 946

21 0.00120195 260.6017 260 53 0.135921 670.8993 670

22 0.639631 914.5046 914 54 0.994646 998.9269 998

23 0.180364 709.9536 709 55 0.951459 990.0976 990

24 0.519327 877.1789 877 56 0.566698 892.6277 892

25 0.0999769 630.9282 630 57 0.900808 979.3241 979

26 0.405242 834.724 834 58 0.630475 911.8714 911

27 0.211588 732.9903 732 59 0.17321 704.2301 704

28 0.271801 770.6376 770 60 0.0132181 420.9532 420

29 0.869522 972.425 972 61 0.837393 965.1301 965

30 0.981092 996.1895 996 62 0.773488 949.9281 949

31 0.00867574 386.9558 386 63 0.435168 846.7036 846

32 0.411859 837.4323 837 64 0.132321 667.3071 667




For the sake of sensible visualization, the processed random numbers in Table 8 are further

arranged into an 8 by 8 arrays as follows:

933 783 994 630 992 911 823 979

736 982 690 834 746 759 819 911

966 789 831 732 998 830 784 704

844 709 781 770 758 470 946 420

980 869 260 972 984 837 670 965

946 760 914 996 573 896 998 949

922 995 709 386 978 950 990 846

818 673 877 837 967 961 892 667

Based on the 8 by 8 arrays listed above, the number of 3-digit number with three

different digits is 45. In order to determine whether the probability that a number has three

different digits is more than the probability suggested in Section 2.5, which is p = 0.73,

hypothesis testing at significance levels of 10% and 5% are conducted.

If H0 is true, then p = 0.73. So, X ~ B(64, 0.73). Since n = 64 (> 30), np = 64 × 0.73 =

46.72 (> 5) and nq = 64 × 0.27 = 17.28, hence normal approximation to the binomial

distribution is employed with a continuity correction factor of 0.5.

Thus, X = 44.5 and

X ~ N(64 × 0.73, 64 × 0.73 × 0.27)

X ~ N(46.72, 12.61)

Then, 46.72

12.61

XZ

Test statistic: 44.5 46.72

0.625212.61

Z

For hypothesis test with significance level of 5%, the cut-off region is at z = 1.645.

But, based on the test statistic, since the Zobserved = -0.6252 (< 1.645), the sample value 45 is

not in the critical region. Thus, it can be concluded that the suggested probability is true.

Likewise, for the test of hypothesis at significance level of 10%, the upper critical

point in such a way that the null hypothesis could be rejected is 1.281. Again, Zobserved < 1.281,

therefore, it can be concluded that the statement on the probability suggested is justified.



CONCLUSIONS 17

4.0 CONCLUSIONS

The studies conducted above clearly manifested that subjective do exists in any

applications in real-life situations, since it is significantly affect by past experiences and one‟s

degree of belief. Although its determination do not possesses any standard mathematical

formulation and equations, but its outcome is often validated by the aid of inferential statistics

in which interval estimations and hypothesis testing are conducted, and inferences are made.

According to statistical analyses conducted on various samples obtained above, it can be

concluded that the probability that a 3-digit number has three distinct digit is the highest

which lies between 0.65 < P(X = 0) < 0.75. This is true as if a census is conducted on each

data in a population of one thousand 3-digit numbers ranged from 0 to 999; the probability of

getting a 3-digit number with three different digits is also lies in the ranges from 0.65 to 0.75.

Besides, the results obtained from testing of hypotheses by Chi-square and one-tail tests also

lead to the probabilities lie between the confidence limits as computed in Section 3.2. As a

verdict, it can be concluded that the inferences and conclusions drawn from the statistical

analyses on the distribution of digit in random numbers are accepted and properly justified.



REFERENCES 18

5.0 REFERENCES

Anscombe, FJ & Aumann, RJ 1963, „A Definition of Subjective Probability‟, The Annals of

Mathematical Statistics, vol. 34, No. 1, pp. 199-205, JSTOR, viewed 9 September

2013.

Bafna, A & Kumar, S 2012, „ProActive Approach for Generating Random Passwords for

Information Protection‟, 2nd

International Conference on Computer, Communication,

Control and Information Technology (C31T-2012), vol. 4, pp. 129-133.

Benham, PP, Crawford, RJ & Armstrong, CG 1987, Mechanics of Engineering Materials, 2nd

edn., Pearson Longman Group Limited, China.

Biedermann, A, Garbolino, P & Taroni, F 2013, „The subjectivist interpretation of probability

and the problem of individualization in forensic science‟, Science & Justice, vol. 53,

pp. 192-200, Elsevier ScienceDirect, viewed 9 September 2013.

Blackjack Info 2013, Blackjack Dealer Outcome Probabilities, BlackjackInfo.com

<www.blackjackinfo.com/bjtourn-dealercharts.php>, viewed 9 September 2013.

Chai, A 2012, Decision Making in HES3360 Human Factors [slide], Swinburne University of

Technology (Sarawak Campus), Kuching, Sarawak.

Christenstock 2013, Blackjack – Using the Probability Theory to Increase your Odds,

Hubpages Inc., <christenstock.hubpages.com/hub/BlackJack---Using-The-Probability-

Theory>, viewed 9 September 2013.

Clemen, RT 1997, CHAPTER 8 – Subjective Probability in Making Hard Decisions: An

Introduction to Decision Analysis (Business Statistics), 2nd

edn., Duxbury.

Crawshaw, J & Chambers, J 2002, A CONCISE COURSE IN ADVANCED LEVEL

STATISTICS With Worked Examples, 4th

edn., Nelson Thones Ltd., United Kingdom.

Elsevier Inc. 2012, CHAPTER 11 – RANDOM NUMBERS in Probability and Random

Processes, Academic Press.

Hibbeler, RC 2009, Structural Analysis, 7th

edn., Pearson Prentice Hall, Singapore.

Jacob, SM 2011, Probability and Statistics: Sampling and Estimation in HMS211 Engineering

Mathematics 3A [slide], Swinburne University of Technology (Sarawak Campus),

Kuching, Sarawak.

Johnson, R 2005, Chapter 6 – Sampling Distribution in Miller & Freund’s Probability and

Statistics for Engineers, 7th

edn., Pearson Prentice Hall, United States of America.

Kahneman, D & Tversky, A 1972, „Subjective Probability: A Judgement of

Representativeness‟, Cognitive Psychology, vol. 3, pp. 430-454.



REFERENCES 19

Kyburg, H 1978, „SUBJECTIVE PROBABILITY: CRITICISMS, REFLECTIONS, AND

PROBLEMS‟, Journal of Philosophical Logic, vol. 7, pp. 157-180, D. Reidel

Publishing Company, Dordrecht, Holland.

Machina, MJ & Schmeidler, D 1992, „A MORE ROBUST DEFINITION OF SUBJECTIVE

PROBABILITY‟, Econometrica, vol. 60, pp. 745-780.

Marsaglia, G & Zaman, A 1993, „Monkey Tests for Random Number Generators‟, Applied

Computers Mathematics, vol. 26, pp. 1-10, Elsevier ScienceDirect, viewed 9

September 2013.

Maskalevich, T 2011, Probability involving sampling without replacement and dependent

trials in Math 728 Lesson Plan.

Soon, CL & Lau, TK 2013, CHAPTER 16 – Sampling and Estimation in Pre-U Text STPM

Mathematics (T) Third Term, Pearson Malaysia Sdn. Bhd.

Thomson, WT & Dahleh, MD 1998, Theory of Vibration with Applications, 5th

edn., Pearson

Prentice-Hall, Inc., United States of America.

Velasco, JD 2012, „Objective and subjective probability in gene expression‟, Progress in

Biophysics and Molecular Biology, vol. 110, pp. 5-10, Elsevier ScienceDirect, viewed

9 September 2013.

Wallsten, TS & Diederich, A 2001, „Understanding pooled subjective probability estimates‟,

Mathematical Social Sciences, vol. 41, pp. 1-18, Elsevier ScienceDirect, viewed 9

September 2013.

Weir, BS 2013, DNA – Statistical Probability in Encyclopedia of Forensic Sciences, 2nd

edn.,

University of Washington, Seattle, WA, USA, Elsevier Ltd.

Whitlow, R 2000, Basic Soil Mechanics, 4th

edn., Pearson Prentice Hall, Malaysia.

Wolfram 2013, Why Mathematica? Wolfram Mathematica 9, Wolfram Inc.,

<www.wolfram.com>, viewed 13 September 2013.

Wong, BT 2012, Surface Resistance Part II in HES5340 Fluid Mechanics 2 [slide],

Swinburne University of Technology (Sarawak Campus), Kuching, Sarawak.



APPENDICES A1

APPENDICES

Appendix 1 – Computational Codes

1. The computational code used to generate one hundred 3-digit random numbers ranged

from 0 to 999 in Wolfram Mathematica 7.0 is:

RandomInteger[999,100]

The left-hand side of the comma in the square bracket represents the domain, which is

ranged from 0 to 999 in this work, while the right-hand side of the comma is the

number of random integers going to be generated which set to be 100. The general

code is: RandomInteger[domain,n]

2. The computational code used to generate sixty four random numbers ranged from 0 to

999 in Wolfram Mathematica 7.0 is:

RandomReal[1,64]

3. The integer part of the sixty four real numbers generated by the function

51000f n n are obtained by the following formula in Microsoft Excel:

int f n



APPENDICES A2

Appendix 2 – List of Statistical Tables

Table 9: Random Number Tables



APPENDICES A3



APPENDICES A4



APPENDICES A5



APPENDICES A6



APPENDICES A7

Table 10: Percentage Points of the 2 Distribution



APPENDICES A8



APPENDICES A9

Table 11: The Upper Tail Probabilities z of the Standard Normal Distribution X~N(0, 1)



APPENDICES A10



APPENDICES A11

Appendix 3 – Screenshots of Random Numbers Generated from Wolfram Mathematica

7.0

Figure 4: The screenshot of random generation of one hundred 3-digit numbers ranged from 0

to 999 by Wolfram Mathematica 7.0

Figure 5: The screenshot of random generation of sixty four real numbers ranged from 0 to 1

by Wolfram Mathematica 7.0

STPM MATHEMATICS (T) ASSIGNMENT C 2012-2013 Statistical Inferences on the Distribution of Digit in Random Numbers by Stephen, P. Y. Bong

Documents

digit numbers

digit random numbers

distribution of digit

random number tables

french deck

result analysis

bong email

list of tables