Mathematics AI SL
Sample Internal Assessment

Table of content
Rationale
Aim
Research question
Background information
Hypothesis
Data collection
Evaluation of hypothesis
Conclusion
Bibliography

# Investigation on the Variation of Probability of Getting the Maximum Share of a Red Packet in WeChat at a particular drawing based on five different raw data sets (each of width 30)

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## Rationale

“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.

## Aim

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.

## Research question

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)?

## Background information

### What is red envelope in reference with wechat

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.

### What is the procedure of distribution of share in red envelope?

As discussed in the previous sub-heading, the distribution of share is determined by an algorithm. The amount of money disbursed in each drawing ranges widely. There is a coefficient of maximum share which determines the range of money which can be disbursed in each drawing. Usually, this value is equal to 2.1. Thus, in case of each drawing, the amount of money to be disbursed ranges between 0.01 CNY and average residual amount of money times the coefficient of maximum disbursement. The average residual amount of money is defined as follows:

Average Residual = $$\frac{Residual\ Amount\ of\ Money}{Residual\ Time\ of\ Drawings}$$

### What are the money limits per envelope and the number of members allowed to draw from each envelope?

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.

### Basic concept of probability used in this IA

The probability of occurrence of an event is:

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

### Regression correlation coefficient

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 (x1,y1), (x2,y2), (xn,yn) 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, y is the value of dependent variable of each observation, xy is the value of the product of the independent and the dependent variable of each observation, n is the number of observation and denotes the sum of all the observation of the mentioned variable.

By squaring the value of r, the value of the regression coefficient (r2) will be achieved. The value of r2 lies between 0 and 1 where 1 signifies maximum correlation whereas 0 signifies null correlation.

### Chi squared test

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:

X2 value = Σ $$\frac{(O_i-E_i)^2}{E_i}$$

Here, Oi is the observed value, Ei is the expected value, denotes the sum of all the observation of the mentioned variable.

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
Figure 1

### Python programming – a very brief idea

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.

## Hypothesis

### Null hypothesis

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

### Alternate hypothesis

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

## Data collection

### Source of data

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.

### Raw data table

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
Figure 2 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 30 CNY
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
Figure 3 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 60 CNY
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
Figure 4 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 90 CNY
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
Figure 5 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 120 CNY
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
Figure 6 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 150 CNY

### Processed data table

Figure 7 - Table On Processed Data Table For Disbursement Of Money When Total Amount Of Money Is 30 CNY
Figure 8 - Table On Processed Data Table For Disbursement Of Money When Total Amount Of Money Is 60 CNY
Figure 9 - Table On Processed Data Table For Disbursement Of Money When Total Amount Of Money Is 90 CNY
Figure 10 - Table On Processed Data Table For Disbursement Of Money When Total Amount Of Money Is 120 CNY
Figure 11 - Table On Processed Data Table For Disbursement Of Money When Total Amount Of Money Is 150 CNY

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

### Analysis of processed data

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.

### Graphical analysis

Figure 12 - Amount Of Money Disbursed In CNY With Respect To Number Of Term When Total Amount Of Money Is 30 CNY
Figure 13 - Amount Of Money Disbursed In CNY With Respect To Number Of Term When Total Amount Of Money Is 60 CNY
Figure 14 - Amount Of Money Disbursed In CNY With Respect To Number Of Term When Total Amount Of Money Is 90 CNY
Figure 15 - Amount Of Money Disbursed In CNY With Respect To Number Of Term When Total Amount Of Money Is 120 CNY
Figure 16 - Amount Of Money Disbursed In CNY With Respect To Number Of Term When Total Amount Of Money Is 150 CNY

### Choice of axis

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).

### Equation of trendline

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 = -0.0201x + 1.3119

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

y = 0.0171x + 1.735

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

y = 0.0599x + 2.0722

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

y = 0.1058x + 2.3594

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

y = -0.0315x + 5.4877

### Outliers

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.

### Analysis

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 8th 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 22nd 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 27th 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 25th 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 20th term, the maximum share was disbursed.

## Evaluation of hypothesis

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

Figure 17 - Table On Observation Table For Evaluation Of X2

Figure 18 - Table On X2

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

Figure 19 - Table On Evaluation Of X2

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

### Calculation of degree of freedom

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

= 5 - 130 - 1 = 4 × 29 = 116

### Evaluation

Examining the value of X2 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,

### Process of calculation using probability

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
Figure 20 - Table On Processed Data Table For Calculation Of Probability

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
Figure 21 - Table On Processed Data Table For Calculation Of Probability

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
Figure 22 - Table On Processed Data Table For Calculation Of Probability

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
Figure 23 - Table On Processed Data Table For Calculation Of Probability

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
Figure 24 - Table On Processed Data Table For Calculation Of Probability

### Sample calculation

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

### Analysis

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.

## Conclusion

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 = -0.0201x + 1.3119

y = 0.0171x + 1.735

y = 0.0599x + 2.0722

y = 0.1058x + 2.3594

y = -0.0315x + 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 X2 in the test was approximately 112. By relating the value with the table written in the Background Information Section, it can be concluded that Null Hypothesis exists, i.e., no correlation is present between the number of term and the amount of money disbursed. Thus, it can be concluded that amount of money disbursed in each term is operated in a random basis. Thus, prediction of number of term for maximum share is not possible.
• 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.

## Bibliography

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