Mathematics AI SL

Sample Internal Assessment

Table of content

Rationale

Aim

Research question

Background information

Hypothesis

Data collection

Evaluation of hypothesis

Conclusion

Bibliography

11 mins Read

2,080 Words

“Money won is twice as sweet as money earned.” A well-known proverb used in the gambling stands for the fact that the money earned in gambling or by the luck feels more exciting and exaggerating than the hard-earned money of business or service. In our last family trip to Hong Kong during the Chinese New Year, we have planned a stay in Macao for two nights. It was my first interaction with gambling when we visited the biggest casino in China – The Venetian Macao. I came across the power of gambling which may either take one to the height of lifestyle, or to the world of depression due to huge of debts. However, the game of roulette has significantly drawn my attention. It was my belief that there is a scientific approach, if followed, may lead to winning the game.

After my holiday, I went deeper into procedure of the game – Roulette. After studying about the game from various articles, research projects that were previously carried on prediction of correct calls, my interest took a shape of a project. It was in 2017, when I was introduced with concepts and applications of probability in mathematics. It has a significantly contribution behind the mathematical exploration of the analysis on determination of corrected calls in Roulette. My project went well but the Future Prospect of the project was analysis on Red Envelope of WeChat.

This was the moment when I came across the concept of Red Envelope. However, due to some circumstances further work was not carried on at that time. But recently my interest developed again on this field.

I have done a thorough research on the feature of Red Envelope on WeChat. I have gone through a number of research articles and journals where I learnt the algorithm which is followed by WeChat to offer the sum of money in each drawings of Red Envelope. Moreover, I have analyzed several data sets on disbursing cash in each drawing from several newspapers and news articles. However, the answer to the question which was partially answered in the last project on Roulette, was not answered this time, in case of Red Envelope – which term will disburse the maximum amount of money?

Heaped with worries, I decided to research and find the answer to my query. This IA is about the same.

The main motive of this exploration is to determine the variation in probability of determining the maximum share drawn at each drawing. This is to determine a correlation between the number of drawing from the Red Envelope and the amount drawn in each drawing so that any relationship can be derived which may lead to determination of term offering the maximum share.

What is the variation in probability of determining the maximum share of a Red Packet in WeChat at a particular drawing based on five different raw data sets (each of width 30)?

Red Envelope is a feature offered in Chinese multipurpose, social media, messaging and mobile payment app made by Tencent – WeChat. This feature acts as a metaphor to the famous tradition in China – gifting red envelope during any occasion, specially, Chinese New Year. Each envelope usually contains some amount of money which is gifted to the friends and family members as their love, affection, relationship and often as a vode of thanks. WeChat added a feature naming ** Red Envelope** which offers the same, virtually.

Here, a person can send a red envelope with a fixed amount of money in Chinese Yuan (CNY) in a WeChat group. The app gives the user, the liberty to set his desired amount of money and the number of drawings that could be made. Once the settings are done and the envelope is sent, the other members of the group will be able to draw from the envelope. It should be noted that the money disbursed in each drawing is not pre-determined and works on an algorithm. Neither the user who sent the envelope, nor the other members of the group can pre-determine the amount of money disbursed in each drawing.

Once the number of drawings set by the user is reached, no more members can draw money from that red envelope and the envelope will be terminated. The money received by the members will be directly deposited in their bank accounts linked with WeChat. It should be noted that the total amount of money sent in the envelope will be disbursed only if the number of drawings is reached. The envelope remains valid only for a day. Thus, if the total number of drawings is not reached by the end of the day, the remaining amount of money is refunded to the user.

The maximum amount of money that could be added in a red envelope is 200 CNY and the maximum number of drawings could be set to 100.

The probability of occurrence of an event is:

Probability (P) = \(\frac{Number\ of\ Favourable\ Outcome}{Total\ Number\ of\ Sample\ Spaces}\)

Regression correlation coefficient is a tool to measure the strength of the correlation between the independent variable and the dependent variable. The set of values (*x*_{1},*y*_{1}), (*x*_{2},*y*_{2}), (*x*_{n},*y*_{n}) are used to find the value of r as stated by the formula below:

*r* = \(\frac{n\big(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-\big(\sum x\big)^2][n\sum y^2-\big(\sum y\big)^2]}}\)

In the above-mentioned formula,** *** x *is the value of independent variable of each observation,

By squaring the value of ** r**, the value of the regression coefficient (

Chi squared test is a kind of analysis which predicts the existence of any correlation between an independent variable and a dependent variable. The Chi squared value of any given set of data is firstly calculated. Now, based on the type of data, for example, paired data or independent data, the Chi squared value is checked in the Chi squared table which further predicts the existence of any correlation.

The formula of Chi squared value is given below:

*X*^{2} *value =* Σ* *\(\frac{(O_i-E_i)^2}{E_i}\)

Here, *O _{i} *is the observed value,

Now, the Chi squared value is checked in Chi squared table which predicts the existence of any correlation. The Chi squared table is shown below:

df

0.995

0.99

0.975

0.95

0.90

0.10

0.05

0.025

0.01

0.005

**1**

---

---

0.001

0.004

0.016

2.706

3.841

5.024

6.635

7.879

**2**

0.010

0.020

0.051

0.103

0.211

4.605

5.991

7.378

9.210

10.597

**3**

0.072

0.115

0.216

0.352

0.584

6.251

7.815

9.348

11.345

12.838

**4**

0.207

0.297

0.484

0.711

1.064

7.779

9.488

11.143

13.277

14.860

**5**

0.412

0.554

0.831

1.145

1.610

9.236

11.070

12.833

15.086

16.750

**6**

0.676

0.872

1.237

1.635

2.204

10.645

12.592

14.449

16.812

18.548

**7**

0.989

1.239

1.690

2.167

2.833

12.017

14.067

16.013

18.475

20.278

**8**

1.344

1.646

2.180

2.733

3.490

13.362

15.507

17.535

20.090

21.955

**9**

1.735

2.088

2.700

3.325

4.168

14.684

16.919

19.023

21.666

23.589

**10**

2.156

2.558

3.247

3.940

4.865

15.987

18.307

20.483

23.209

25.188

**11**

2.603

3.053

3.816

4.575

5.578

17.275

19.675

21.920

24.725

26.757

**12**

3.074

3.571

4.404

5.226

6.304

18.549

21.026

23.337

26.217

28.300

**13**

3.565

4.107

5.009

5.892

7.042

19.812

22.362

24.736

27.688

29.819

**14**

4.075

4.660

5.629

6.571

7.790

21.064

23.685

26.119

29.141

31.319

**15**

4.601

5.229

6.262

7.261

8.547

22.307

24.996

27.488

30.578

32.801

**16**

5.142

5.812

6.908

7.962

9.312

23.542

26.296

28.845

32.000

34.267

**17**

5.697

6.408

7.564

8.672

10.085

24.769

27.587

30.191

33.409

35.718

**18**

6.265

7.015

8.231

9.390

10.865

25.989

28.869

31.526

34.805

37.156

**19**

6.844

7.633

8.907

10.117

11.651

27.204

30.144

32.852

36.191

38.582

**20**

7.434

8.260

9.591

10.851

12.443

28.412

31.410

34.170

37.566

39.997

**21**

8.034

8.897

10.283

11.591

13.240

29.615

32.671

35.479

38.932

41.401

**22**

8.643

9.542

10.982

12.338

14.041

30.813

33.924

36.781

40.289

42.796

**23**

9.260

10.196

11.689

13.091

14.848

32.007

35.172

38.076

41.638

44.181

**24**

9.886

10.856

12.401

13.848

15.659

33.196

36.415

39.364

42.980

45.559

**25**

10.520

11.524

13.120

14.611

16.473

34.382

37.652

40.646

44.314

46.928

**26**

11.160

12.198

13.844

15.379

17.292

35.563

38.885

41.923

45.642

48.290

**27**

11.808

12.879

14.573

16.151

18.114

36.741

40.113

43.195

46.963

49.645

**28**

12.461

13.565

15.308

16.928

18.939

37.916

41.337

44.461

48.278

50.993

**29**

13.121

14.256

16.047

17.708

19.768

39.087

42.557

45.722

49.588

52.336

**30**

13.787

14.953

16.791

18.493

20.599

40.256

43.773

46.979

50.892

53.672

**40**

20.707

22.164

24.433

26.509

29.051

51.805

55.758

59.342

63.691

66.766

**50**

27.991

29.707

32.357

34.764

37.689

63.167

67.505

71.420

76.154

79.490

**60**

35.534

37.485

40.482

43.188

46.459

74.397

79.082

83.298

88.379

91.952

**70**

43.275

45.442

48.758

51.739

55.329

85.527

90.531

95.023

100.425

104.215

**80**

51.172

53.540

57.153

60.391

64.278

96.578

101.879

106.629

112.329

116.321

**90**

59.196

61.754

65.647

69.126

73.291

107.565

113.145

118.136

124.116

128.299

**100**

67.328

70.065

74.222

77.929

82.358

118.498

124.342

129.561

135.807

140.169

Python is a high-level programming language which is used to serve several purposes in the domain of information technology. In context with this exploration, python programming language can be used to develop a prototype of the feature of red envelope only with respect to the amount of money that should be disbursed.

It is assumed that there does not exist any correlation between the number of drawing and probability of getting the maximum share.

It is assumed that there exists a correlation between the number of drawing and probability of getting the maximum share.

A data sheet has been prepared based on several news articles, reports and surveys in money disbursed in each drawing in Red Envelope WeChat. It has been possible to record the data of number of drawing and amount as these amounts directly reflect in bank statement.

**Justification of the Source and Interval of Raw Data**

Data sheet has been prepared based on the amount of money added in the red envelope by the contributor. In all the data sets, the number of drawings is set to 30. This is treated as a controlled variable to keep a uniformity to study the correlation. The amount of money added in each trial is increased linearly at an interval of 30. This is done to ignore more complex calculations as the number of drawings is kept fixed to 30.

Term

Amount of Money disbursed (in CNY)

1

1.23

2

1.43

3

0.97

4

0.45

5

0.78

6

0.56

7

1.23

8

4.51

9

0.98

10

0.23

11

0.34

12

0.21

13

2.34

14

0.98

15

1.00

16

0.65

17

1.02

18

0.56

19

0.98

20

0.34

21

4.23

22

0.34

23

0.16

24

0.54

25

1.02

26

0.94

27

0.56

28

0.45

29

0.45

30

0.52

Term

Amount of Money disbursed (in CNY)

1

0.65

2

1.65

3

2.34

4

2.65

5

1.8

6

2.00

7

1.78

8

1.43

9

1.76

10

3.54

11

1.34

12

0.65

13

0.34

14

1.23

15

1.65

16

2.43

17

1.43

18

1.76

19

2.34

20

2.98

21

2.34

22

7.65

23

2.43

24

2.98

25

2.34

26

1.54

27

2.00

28

1.23

29

1.18

30

0.56

Term

Amount of Money disbursed (in CNY)

1

3

2

3.23

3

2.34

4

2.65

5

3.87

6

3.45

7

2.12

8

1.54

9

3.76

10

3.98

11

3.56

12

1.23

13

0.45

14

0.34

15

1.54

16

2.34

17

1.23

18

3.45

19

3

20

1.32

21

1.54

22

3.65

23

3.87

24

2.56

25

2.54

26

10.43

27

10.54

28

2.34

29

1.65

30

2.48

Term

Amount of Money disbursed (in CNY)

1

3.54

2

1.76

3

4.65

4

7.65

5

5.43

6

3.34

7

2.43

8

2.64

9

2.76

10

3.54

11

2.43

12

2.98

13

1.65

14

1.65

15

1.54

16

0.98

17

0.76

18

2.54

19

2.86

20

1.54

21

3.65

22

4.87

23

4.65

24

8.76

25

15.43

26

2.54

27

10.98

28

3.54

29

4.54

30

4.37

Term

Amount of Money disbursed (in CNY)

1

5.03

2

5.02

3

5.76

4

4.36

5

5.03

6

5.32

7

5.65

8

4.87

9

5.34

10

5.76

11

4.67

12

5.09

13

5.87

14

4.56

15

4.56

16

4.87

17

5.67

18

5.23

19

5.67

20

5.98

21

4.87

22

4.67

23

4.56

24

4.98

25

5.87

26

5.89

27

5.03

28

4.45

29

4.34

30

1.03

*Sample Calculation:*

Mean = \(\frac{y_1+y_2+...+y_n}{n}\)

Arithmetic Mean = \(\frac{1.23+1.43+0.97+…+0.94+0.56}{30}\) = 1.00

Standard Deviation = \(\frac{\sqrt{(\bar y-y_1)^2+(\bar y-y_2)^2+...+(\bar y-y_n)^2}}{n}\)

Standard Deviation = \(\frac{\sqrt{(1-1.23)^2+(1-1.43)^2+…+(1-0.56)^2}}{30}\) = 1.02

From the processed data table, the mean and standard deviation of each data set has been calculated. From the first data set, it has been found that the mean is 1 CNY. The standard deviation of the data set is found to be minimum amongst the other data sets which is equal to 1.02. From the second data set, it has been found that the mean is 2 CNY. The standard deviation of the data set is found to be minimum amongst the other data sets which is equal to 1.30. From the third data set, it has been found that the mean is 3 CNY. The standard deviation of the data set is found to be minimum amongst the other data sets which is equal to 2.27. From the fourth data set, it has been found that the mean is 4 CNY. The standard deviation of the data set is found to be minimum amongst the other data sets which is equal to 3.12. From the fifth data set, it has been found that the mean is 5 CNY. The standard deviation of the data set is found to be minimum amongst the other data sets which is equal to 0.90.

As the standard deviation in each of the data set is not exactly equal to zero, this table will not have a significant contribution in analyzing the maximum share as a minute difference in share will result in determination of maximum and minimum disbursement.

The X – Axis of the graph denotes the number of term or number of drawing of money from the red envelope (independent variable).

The Y – Axis of the graph denotes the amount of money disbursed in CNY in each drawing (dependent variable).

In this graph, a linear trendline has been obtained using the data that has been collected based on the survey done in several newspaper and news articles. The equation of the trendline for first data set (30 CNY) is shown below:

*y* = -0.0201*x* + 1.3119

The equation of the trendline for second data set (60 CNY) is shown below:

*y* = 0.0171*x *+ 1.735

The equation of the trendline for third data set (90 CNY) is shown below:

*y* = 0.0599*x* + 2.0722

The equation of the trendline for fourth data set (120 CNY) is shown below:

*y* = 0.1058*x* + 2.3594

The equation of the trendline for fifth data set (150 CNY) is shown below:

*y* = -0.0315*x* + 5.4877

There are several outliers observed in the graph obtained by plotting the values of the dataset. The prime reason behind the presence of outliers is the algorithm which is followed to coin the amount of money disbursed. There is no definitely correlation between the number of terms and amount of money disbursed as the correlation coefficient obtained in each of the graphs are very close to zero. Thus, there are a lot of outliers in the graphs are obtained.

For Graph 1:

There are a lot of outliers of which one is also the maximum share disbursed in the graph. A linear trendline is obtained based on the dataset but the regression correlation coefficient is 0.03. Thus, it can be concluded that the correlation does not exist. However, according to the correlation obtained, there exist a decreasing relationship between the number of term and amount of money disbursed. In a contrary, in 8^{th} term, the maximum share was disbursed.

For Graph 2:

There are a lot of outliers of which one is also the maximum share disbursed in the graph. A linear trendline is obtained based on the dataset but the regression correlation coefficient is 0.01. Thus, it can be concluded that the correlation does not exist. However, according to the correlation obtained, there exist an increasing relationship between the number of term and amount of money disbursed. In a contrary, in 22^{nd} term, the maximum share was disbursed.

For Graph 3:

There are a lot of outliers of which one is also the maximum share disbursed in the graph. A linear trendline is obtained based on the dataset but the regression correlation coefficient is 0.05. Thus, it can be concluded that the correlation does not exist. However, according to the correlation obtained, there exist an increasing relationship between the number of term and amount of money disbursed. In a contrary, in 27^{th} term, the maximum share was disbursed.

For Graph 4:

There are a lot of outliers of which one is also the maximum share disbursed in the graph. A linear trendline is obtained based on the dataset but the regression correlation coefficient is 0.08. Thus, it can be concluded that the correlation does not exist. However, according to the correlation obtained, there exist an increasing relationship between the number of term and amount of money disbursed. In a contrary, in 25^{th} term, the maximum share was disbursed.

For Graph 5:

There are a lot of outliers of which one is also the maximum share disbursed in the graph. A linear trendline is obtained based on the dataset but the regression correlation coefficient is 0.09. Thus, it can be concluded that the correlation does not exist. However, according to the correlation obtained, there exist a decreasing relationship between the number of term and amount of money disbursed. In a contrary, in 20^{th} term, the maximum share was disbursed.

The hypothesis has been evaluated with the help of *X *^{2 }– Test in this section of this mathematical exploration. The *X *^{2 }– Test will conclude whether or not the null hypothesis or the alternate hypothesis is true.

Observed Value (O)

Expected Value (E)

(O - E)

(O - E)^{2}

\(\frac{(O-E)^2}{E}\)

1.23

0.897

0.333

0.110889

0.12362207

1.43

0.873

0.557

0.310249

0.35538259

0.97

1.071

-0.101

0.010201

0.00952474

0.45

1.184

-0.734

0.538756

0.45503041

0.78

1.127

-0.347

0.120409

0.10684028

0.56

0.978

-0.418

0.174724

0.1786544

1.23

0.881

0.349

0.121801

0.13825312

4.51

0.999

3.511

12.327121

12.3394605

0.98

0.973

0.007

4.9E-05

5.036E-05

0.23

1.137

-0.907

0.822649

0.72352595

0.34

0.823

-0.483

0.233289

0.28346173

0.21

0.677

-0.467

0.218089

0.32214032

2.34

0.71

1.63

2.6569

3.74211268

0.98

0.584

0.396

0.156816

0.26852055

1

0.686

0.314

0.098596

0.14372595

0.65

0.751

-0.101

0.010201

0.01358322

1.02

0.674

0.346

0.119716

0.17762018

0.56

0.903

-0.343

0.117649

0.13028682

0.98

0.99

-0.01

0.0001

0.00010101

0.34

0.811

-0.471

0.221841

0.27354007

4.23

1.109

3.121

9.740641

8.7832651

0.34

1.412

-1.072

1.149184

0.81386969

0.16

1.045

-0.885

0.783225

0.74949761

0.54

1.321

-0.781

0.609961

0.46174186

1.02

1.813

-0.793

0.628849

0.34685549

0.94

1.423

-0.483

0.233289

0.16394167

0.56

1.941

-1.381

1.907161

0.9825662

0.45

0.801

-0.351

0.123201

0.15380899

0.45

0.811

-0.361

0.130321

0.16069174

0.52

0.597

-0.077

0.005929

0.00993132

0.65

1.793

-1.143

1.306449

0.72863859

1.65

1.745

-0.095

0.009025

0.00517192

2.34

2.141

0.199

0.039601

0.0184965

2.65

2.368

0.282

0.079524

0.03358277

1.8

2.255

-0.455

0.207025

0.0918071

2

1.956

0.044

0.001936

0.00098978

1.78

1.761

0.019

0.000361

0.000205

1.43

1.999

-0.569

0.323761

0.16196148

1.76

1.947

-0.187

0.034969

0.01796045

3.54

2.273

1.267

1.605289

0.70624241

1.34

1.645

-0.305

0.093025

0.05655015

0.65

1.355

-0.705

0.497025

0.36680812

0.34

1.42

-1.08

1.1664

0.82140845

1.23

1.168

0.062

0.003844

0.0032911

1.65

1.372

0.278

0.077284

0.05632945

2.43

1.503

0.927

0.859329

0.57174251

1.43

1.348

0.082

0.006724

0.00498813

1.76

1.805

-0.045

0.002025

0.00112188

2.34

1.98

0.36

0.1296

0.06545455

2.98

1.621

1.359

1.846881

1.1393467

2.34

2.217

0.123

0.015129

0.00682409

7.65

2.824

4.826

23.290276

8.24726487

2.43

2.089

0.341

0.116281

0.05566348

2.98

2.643

0.337

0.113569

0.04296973

2.34

3.627

-1.287

1.656369

0.45667742

1.54

2.845

-1.305

1.703025

0.59860281

2

3.881

-1.881

3.538161

0.9116622

1.23

1.601

-0.371

0.137641

0.08597189

1.18

1.621

-0.441

0.194481

0.11997594

0.56

1.195

-0.635

0.403225

0.33742678

3

2.69

0.31

0.0961

0.03572491

3.23

2.618

0.612

0.374544

0.14306494

2.34

3.212

-0.872

0.760384

0.23673225

2.65

3.552

-0.902

0.813604

0.22905518

3.87

3.382

0.488

0.238144

0.07041514

3.45

2.934

0.516

0.266256

0.09074847

2.12

2.642

-0.522

0.272484

0.1031355

1.54

2.998

-1.458

2.125764

0.70906071

3.76

2.92

0.84

0.7056

0.24164384

3.98

3.41

0.57

0.3249

0.09527859

3.56

2.468

1.092

1.192464

0.48317018

1.23

2.032

-0.802

0.643204

0.3165374

0.45

2.13

-1.68

2.8224

1.32507042

0.34

1.752

-1.412

1.993744

1.13798174

1.54

2.058

-0.518

0.268324

0.13038095

2.34

2.254

0.086

0.007396

0.00328128

1.23

2.022

-0.792

0.627264

0.31021958

3.45

2.708

0.742

0.550564

0.20331019

3

2.97

0.03

0.0009

0.00030303

1.32

2.432

-1.112

1.236544

0.50844737

1.54

3.326

-1.786

3.189796

0.95904871

3.65

4.236

-0.586

0.343396

0.0810661

3.87

3.134

0.736

0.541696

0.17284493

2.56

3.964

-1.404

1.971216

0.49727952

2.54

5.44

-2.9

8.41

1.54595588

10.43

4.268

6.162

37.970244

8.89649578

10.54

5.822

4.718

22.259524

3.82334662

2.34

2.402

-0.062

0.003844

0.00160033

1.65

2.432

-0.782

0.611524

0.25144901

2.48

1.792

0.688

0.473344

0.26414286

3.54

3.587

-0.047

0.002209

0.00061583

1.76

3.491

-1.731

2.996361

0.85831023

4.65

4.283

0.367

0.134689

0.03144735

7.65

4.736

2.914

8.491396

1.79294679

5.43

4.509

0.921

0.848241

0.18812176

3.34

3.912

-0.572

0.327184

0.08363599

2.43

3.523

-1.093

1.194649

0.33909991

2.64

3.997

-1.357

1.841449

0.46070778

2.76

3.893

-1.133

1.283689

0.32974287

3.54

4.547

-1.007

1.014049

0.22301495

2.43

3.291

-0.861

0.741321

0.22525706

2.98

2.709

0.271

0.073441

0.02711

1.65

2.84

-1.19

1.4161

0.49862676

1.65

2.336

-0.686

0.470596

0.20145377

1.54

2.744

-1.204

1.449616

0.52828571

0.98

3.005

-2.025

4.100625

1.36460067

0.76

2.696

-1.936

3.748096

1.39024332

2.54

3.611

-1.071

1.147041

0.3176519

2.86

3.96

-1.1

1.21

0.30555556

1.54

3.243

-1.703

2.900209

0.89429818

3.65

4.435

-0.785

0.616225

0.13894589

4.87

5.648

-0.778

0.605284

0.10716785

4.65

4.179

0.471

0.221841

0.05308471

8.76

5.285

3.475

12.075625

2.28488647

15.43

7.253

8.177

66.863329

9.2187135

2.54

5.691

-3.151

9.928801

1.74464962

10.98

7.763

3.217

10.349089

1.3331301

3.54

3.203

0.337

0.113569

0.03545707

4.54

3.243

1.297

1.682209

0.51872001

4.37

2.389

1.981

3.924361

1.64267936

5.03

4.483

0.547

0.299209

0.06674303

5.02

4.363

0.657

0.431649

0.09893399

5.76

5.353

0.407

0.165649

0.03094508

4.36

5.92

-1.56

2.4336

0.41108108

5.03

5.637

-0.607

0.368449

0.0653626

5.32

4.89

0.43

0.1849

0.03781186

5.65

4.403

1.247

1.555009

0.35317034

4.87

4.997

-0.127

0.016129

0.00322774

5.34

4.867

0.473

0.223729

0.04596856

5.76

5.683

0.077

0.005929

0.00104329

4.67

4.113

0.557

0.310249

0.07543132

5.09

3.387

1.703

2.900209

0.85627665

5.87

3.55

2.32

5.3824

1.51616901

4.56

2.92

1.64

2.6896

0.92109589

4.56

3.43

1.13

1.2769

0.37227405

4.87

3.757

1.113

1.238769

0.32972292

5.67

3.37

2.3

5.29

1.56973294

5.23

4.513

0.717

0.514089

0.11391292

5.67

4.95

0.72

0.5184

0.10472727

5.98

4.053

1.927

3.713329

0.9161927

4.87

5.543

-0.673

0.452929

0.08171189

4.67

7.06

-2.39

5.7121

0.80907932

4.56

5.223

-0.663

0.439569

0.08416025

4.98

6.607

-1.627

2.647129

0.40065521

5.87

9.067

-3.197

10.220809

1.12725367

5.89

7.113

-1.223

1.495729

0.21028103

5.03

9.703

-4.673

21.836929

2.25053375

4.45

4.003

0.447

0.199809

0.04991481

4.34

4.053

0.287

0.082369

0.02032297

1.03

2.987

-1.957

3.829849

1.28217241

\(\sum\frac{(O-E)^2}{E}\) = 0.123 + 0.355 +…+ 1.282 = 112.33

*Degree of Freedom = *(*Column - 1*)(*Row - 1*)

= 5 - 130 - 1 = 4 × 29 = 116

Examining the value of *X*^{2}* *with respect to the degree of freedom using the table as shown in Background Information Section, it is concluded that the Null Hypothesis is accepted and the Alternate Hypothesis is rejected.

As there is no relation between the number of term and amount of money disbursed,

The concept of Probability provides information about the chances of getting a desired value or event at any particular instant. In this exploration, the concept of Probability will be used to determine the chances of getting the maximum share in a particular term or drawing. For example, if at any definite term of drawing money from the red envelope, there are N number of shares possible and the number of maximum share is 1, then the probability of getting maximum share in that drawing will be \(\frac{1}P\). This will be used in a tabular form in relation with the observed data to analyse the outcome of maximum share.

For Graph 1:

Term

Amount Drawn (CNY)

Residual Drawing (CNY)

Average Residual Amount (CNY)

Maximum Share Possible (CNY)

Probability

0

0

30

1.000

2.10

0.005

1

1.23

28.77

0.992

2.08

0.005

2

1.43

27.34

0.976

2.05

0.005

3

0.97

26.37

0.977

2.05

0.005

4

0.45

25.92

0.997

2.09

0.005

5

0.78

25.14

1.006

2.11

0.005

6

0.56

24.58

1.024

2.15

0.005

7

1.23

23.35

1.015

2.13

0.005

8

4.51

18.84

0.856

1.80

0.006

9

0.98

17.86

0.850

1.79

0.006

10

0.23

17.63

0.882

1.85

0.005

11

0.34

17.29

0.910

1.91

0.005

12

0.21

17.08

0.949

1.99

0.005

13

2.34

14.74

0.867

1.82

0.005

14

0.98

13.76

0.860

1.81

0.006

15

1.00

12.76

0.851

1.79

0.006

16

0.65

12.11

0.865

1.82

0.006

17

1.02

11.09

0.853

1.79

0.006

18

0.56

10.53

0.878

1.84

0.005

19

0.98

9.55

0.868

1.82

0.005

20

0.34

9.21

0.921

1.93

0.005

21

4.23

4.98

0.553

1.16

0.009

22

0.34

4.64

0.580

1.22

0.008

23

0.16

4.48

0.640

1.34

0.007

24

0.54

3.94

0.657

1.38

0.007

25

1.02

2.92

0.584

1.23

0.008

26

0.94

1.98

0.495

1.04

0.010

27

0.56

1.42

0.473

0.99

0.010

28

0.45

0.97

0.485

1.02

0.010

29

0.45

0.52

0.520

1.09

0.009

30

0.52

0

0

0.520

0

For Graph 2:

Term

Amount Drawn (CNY)

Residual Drawing (CNY)

Average Residual Amount (CNY)

Maximum Share Possible (CNY)

Probability

0

0

60

2.00

4.20

0.002

1

0.65

59.35

2.05

4.30

0.002

2

1.65

57.7

2.06

4.33

0.002

3

2.34

55.36

2.05

4.31

0.002

4

2.65

52.71

2.03

4.26

0.002

5

1.8

50.91

2.04

4.28

0.002

6

2

48.91

2.04

4.28

0.002

7

1.78

47.13

2.05

4.30

0.002

8

1.43

45.7

2.08

4.36

0.002

9

1.76

43.94

2.09

4.39

0.002

10

3.54

40.4

2.02

4.24

0.002

11

1.34

39.06

2.06

4.32

0.002

12

0.65

38.41

2.13

4.48

0.002

13

0.34

38.07

2.24

4.70

0.002

14

1.23

36.84

2.30

4.84

0.002

15

1.65

35.19

2.35

4.93

0.002

16

2.43

32.76

2.34

4.91

0.002

17

1.43

31.33

2.41

5.06

0.002

18

1.76

29.57

2.46

5.17

0.002

19

2.34

27.23

2.48

5.20

0.002

20

2.98

24.25

2.43

5.09

0.002

21

2.34

21.91

2.43

5.11

0.002

22

7.65

14.26

1.78

3.74

0.003

23

2.43

11.83

1.69

3.55

0.003

24

2.98

8.85

1.48

3.10

0.003

25

2.34

6.51

1.30

2.73

0.004

26

1.54

4.97

1.24

2.61

0.004

27

2

2.97

0.99

2.08

0.005

28

1.23

1.74

0.87

1.83

0.005

29

1.18

0.56

0.56

1.18

0.009

30

0.56

0

0

0.00

0

For Graph 3:

Term

Amount Drawn (CNY)

Residual Drawing (CNY)

Average Residual Amount (CNY)

Maximum Share Possible (CNY)

Probability

0

0

90

3.000

6.30

0.0016

1

3

87

3.000

6.30

0.0016

2

3.23

83.77

2.992

6.28

0.0016

3

2.34

81.43

3.016

6.33

0.0016

4

2.65

78.78

3.030

6.36

0.0016

5

3.87

74.91

2.996

6.29

0.0016

6

3.45

71.46

2.978

6.25

0.0016

7

2.12

69.34

3.015

6.33

0.0015

8

1.54

67.8

3.082

6.47

0.0016

9

3.76

64.04

3.050

6.40

0.0016

10

3.98

60.06

3.003

6.30

0.0016

11

3.56

56.5

2.974

6.24

0.0016

12

1.23

55.27

3.071

6.44

0.0015

13

0.45

54.82

3.225

6.77

0.0014

14

0.34

54.48

3.405

7.15

0.0013

15

1.54

52.94

3.529

7.41

0.0013

16

2.34

50.6

3.614

7.59

0.0013

17

1.23

49.37

3.798

7.97

0.0012

18

3.45

45.92

3.827

8.03

0.0012

19

3

42.92

3.902

8.19

0.0011

20

1.32

41.6

4.160

8.73

0.0011

21

1.54

40.06

4.451

9.34

0.0010

22

3.65

36.41

4.551

9.55

0.0010

23

3.87

32.54

4.649

9.76

0.0010

24

2.56

29.98

4.997

10.43

0.0009

25

2.54

27.44

5.488

11.55

0.0011

26

10.43

17.01

4.252

8.90

0.0022

27

10.54

6.47

2.157

4.29

0.0023

28

2.34

4.13

2.065

4.36

0.0019

29

1.65

2.48

2.480

5.08

0.0016

30

2.48

0

0

0.00

0.0016

For Graph 4:

Term

Amount Drawn (CNY)

Residual Drawing (CNY)

Average Residual Amount (CNY)

Maximum Share Possible (CNY)

Probability

0

0

120

4

8.4

0.0012

1

3.54

116.46

4.01586207

8.43331034

0.0012

2

1.76

114.7

4.09642857

8.6025

0.0012

3

4.65

110.05

4.07592593

8.55944444

0.0012

4

7.65

102.4

3.93846154

8.27076923

0.0012

5

5.43

96.97

3.8788

8.14548

0.0012

6

3.34

93.63

3.90125

8.192625

0.0012

7

2.43

91.2

3.96521739

8.32695652

0.0012

8

2.64

88.56

4.02545455

8.45345455

0.0012

9

2.76

85.8

4.08571429

8.58

0.0012

10

3.54

82.26

4.113

8.6373

0.0012

11

2.43

79.83

4.20157895

8.82331579

0.0011

12

2.98

76.85

4.26944444

8.96583333

0.0011

13

1.65

75.2

4.42352941

9.28941176

0.0011

14

1.65

73.55

4.596875

9.6534375

0.0010

15

1.54

72.01

4.80066667

10.0814

0.0010

16

0.98

71.03

5.07357143

10.6545

0.0009

17

0.76

70.27

5.40538462

11.3513077

0.0009

18

2.54

67.73

5.64416667

11.85275

0.0008

19

2.86

64.87

5.89727273

12.3842727

0.0008

20

1.54

63.33

6.333

13.2993

0.0008

21

3.65

59.68

6.63111111

13.9253333

0.0007

22

4.87

54.81

6.85125

14.387625

0.0007

23

4.65

50.16

7.16571429

15.048

0.0007

24

8.76

41.4

6.9

14.49

0.0007

25

15.43

25.97

5.194

10.9074

0.0009

26

2.54

23.43

5.8575

12.30075

0.0008

27

10.98

12.45

4.15

8.715

0.0011

28

3.54

8.91

4.455

9.3555

0.0011

29

4.54

4.37

4.37

9.177

0.0011

30

4.37

0

0

0

0.0000

For Graph 5:

Term

Amount Drawn (CNY)

Residual Drawing (CNY)

Average Residual Amount (CNY)

Maximum Share Possible (CNY)

Probability

0

0

150

5.000

10.500

0.00095

1

5.03

144.97

4.999

10.498

0.00095

2

5.02

139.95

4.998

10.496

0.00095

3

5.76

134.19

4.970

10.437

0.00096

4

4.36

129.83

4.993

10.486

0.00095

5

5.03

124.8

4.992

10.483

0.00095

6

5.32

119.48

4.978

10.455

0.00096

7

5.65

113.83

4.949

10.393

0.00096

8

4.87

108.96

4.953

10.401

0.00096

9

5.34

103.62

4.934

10.362

0.00097

10

5.76

97.86

4.893

10.275

0.00097

11

4.67

93.19

4.905

10.300

0.00097

12

5.09

88.1

4.894

10.278

0.00097

13

5.87

82.23

4.837

10.158

0.00098

14

4.56

77.67

4.854

10.194

0.00098

15

4.56

73.11

4.874

10.235

0.00098

16

4.87

68.24

4.874

10.236

0.00098

17

5.67

62.57

4.813

10.107

0.00099

18

5.23

57.34

4.778

10.035

0.00100

19

5.67

51.67

4.697

9.864

0.00101

20

5.98

45.69

4.569

9.595

0.00104

21

4.87

40.82

4.536

9.525

0.00105

22

4.67

36.15

4.519

9.489

0.00105

23

4.56

31.59

4.513

9.477

0.00106

24

4.98

26.61

4.435

9.313

0.00107

25

5.87

20.74

4.148

8.711

0.00115

26

5.89

14.85

3.712

7.796

0.00128

27

5.03

9.82

3.273

6.874

0.00145

28

4.45

5.37

2.685

5.638

0.00177

29

4.34

1.03

1.030

2.163

0.00462

30

1.03

0

0.000

0.000

0.00000

Residual Amount *= Amount in Red Envelope - Amount Drawn *= 150 - 5.03 = 144.97

Average Residual Amount = \(\frac{Residual\ Amount}{Number\ of\ Drawings\ Remaining}\) = \(\frac{144.97}{29}\) = 4.999

Maximum Share = Average Residual Amount × Maximum Share Coefficient = 4.99 × 2.1 = 10.49

Probability of maximum share = \(\frac{1}{Number\ of\ Possible\ Outcome}\) = \(\frac{1}{1049}\) = 0.00095

The probability analysis was not been able to decipher the relationship to predict the term with maximum share disbursement. This is because of several anomalous behavior that has been obtained in the algorithm while the probability analysis was carried on. From the Background Information Section, it was noted that the range of money that could be disbursed is between 0.01 CNY and average residual amount times maximum share coefficient. The maximum share coefficient was assumed to be constant and equal to 2.1. However, there are several instances where the amount of money disbursed is greater than considered range. This states that the maximum share coefficient is not a constant term. It is a variable; however, any information on how it varies is unknown. The probability of getting the maximum share in most of the observation in the upper side of each table approximately equal. Thus, it is not possible to distinctly state the term with maximum share disbursement. On the other hand, there are terms with significantly less probability of getting maximum share than that of the others but those have received the maximum share. Thus, it can be stated that the process of disbursement is completely dynamic and random. Thus, probability analysis cannot serve to purpose of prediction of term with maximum share.

There is no definite relationship or formulation to predict the disbursement of maximum share at any term. Thus, it is concluded that it cannot be predicted beforehand whether or not any term will have the maximum share of disbursement.

- There exist no correlation between the number of term and amount of money disbursed in each drawing.
- The equation of trendline for Group 1 to Group 5 are as follows:

*y* = -0.0201*x* + 1.3119

*y* = 0.0171*x* + 1.735

*y* = 0.0599*x* + 2.0722

*y* = 0.1058*x* + 2.3594

*y *= -0.0315*x* + 5.4877

- The regression correlation coefficient in obtained in each data set are 0.03, 0.01, 0.05, 0.08 and 0.09 for Graphs 1 to 5. As the strength of correlation is very weak, it can be concluded that the linear correlation is invalid.
- The value of
*X*^{2} - The coefficient of maximum share is not constant.
- Amount of money disbursed in each term could be more than upper limit of the range of sum that should be disbursed.
- The probability of getting the maximum share in a term is approximately same for all the terms in each data set. However, the probability is comparatively more in initial drawings and comparatively less in later drawings.
- It should be noted that maximum share is disbursed in a term with less probability than others.

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