Mathematics AI SL's Sample Internal Assessment

Mathematics AI SL's Sample Internal Assessment

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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Word count: 2,080

Table of content

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:

df0.9950.990.9750.950.900.100.050.0250.010.005

1

------0.0010.0040.0162.7063.8415.0246.6357.879

2

0.0100.0200.0510.1030.2114.6055.9917.3789.21010.597

3

0.0720.1150.2160.3520.5846.2517.8159.34811.34512.838

4

0.2070.2970.4840.7111.0647.7799.48811.14313.27714.860

5

0.4120.5540.8311.1451.6109.23611.07012.83315.08616.750

6

0.6760.8721.2371.6352.20410.64512.59214.44916.81218.548

7

0.9891.2391.6902.1672.83312.01714.06716.01318.47520.278

8

1.3441.6462.1802.7333.49013.36215.50717.53520.09021.955

9

1.7352.0882.7003.3254.16814.68416.91919.02321.66623.589

10

2.1562.5583.2473.9404.86515.98718.30720.48323.20925.188

11

2.6033.0533.8164.5755.57817.27519.67521.92024.72526.757

12

3.0743.5714.4045.2266.30418.54921.02623.33726.21728.300

13

3.5654.1075.0095.8927.04219.81222.36224.73627.68829.819

14

4.0754.6605.6296.5717.79021.06423.68526.11929.14131.319

15

4.6015.2296.2627.2618.54722.30724.99627.48830.57832.801

16

5.1425.8126.9087.9629.31223.54226.29628.84532.00034.267

17

5.6976.4087.5648.67210.08524.76927.58730.19133.40935.718

18

6.2657.0158.2319.39010.86525.98928.86931.52634.80537.156

19

6.8447.6338.90710.11711.65127.20430.14432.85236.19138.582

20

7.4348.2609.59110.85112.44328.41231.41034.17037.56639.997

21

8.0348.89710.28311.59113.24029.61532.67135.47938.93241.401

22

8.6439.54210.98212.33814.04130.81333.92436.78140.28942.796

23

9.26010.19611.68913.09114.84832.00735.17238.07641.63844.181

24

9.88610.85612.40113.84815.65933.19636.41539.36442.98045.559

25

10.52011.52413.12014.61116.47334.38237.65240.64644.31446.928

26

11.16012.19813.84415.37917.29235.56338.88541.92345.64248.290

27

11.80812.87914.57316.15118.11436.74140.11343.19546.96349.645

28

12.46113.56515.30816.92818.93937.91641.33744.46148.27850.993

29

13.12114.25616.04717.70819.76839.08742.55745.72249.58852.336

30

13.78714.95316.79118.49320.59940.25643.77346.97950.89253.672

40

20.70722.16424.43326.50929.05151.80555.75859.34263.69166.766

50

27.99129.70732.35734.76437.68963.16767.50571.42076.15479.490

60

35.53437.48540.48243.18846.45974.39779.08283.29888.37991.952

70

43.27545.44248.75851.73955.32985.52790.53195.023100.425104.215

80

51.17253.54057.15360.39164.27896.578101.879106.629112.329116.321

90

59.19661.75465.64769.12673.291107.565113.145118.136124.116128.299

100

67.32870.06574.22277.92982.358118.498124.342129.561135.807140.169

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

TermAmount of Money disbursed (in CNY)
11.23
21.43
30.97
40.45
50.78
60.56
71.23
84.51
90.98
100.23
110.34
120.21
132.34
140.98
151.00
160.65
171.02
180.56
190.98
200.34
214.23
220.34
230.16
240.54
251.02
260.94
270.56
280.45
290.45
300.52

Figure 2 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 30 CNY

TermAmount of Money disbursed (in CNY)
10.65
21.65
32.34
42.65
51.8
62.00
71.78
81.43
91.76
103.54
111.34
120.65
130.34
141.23
151.65
162.43
171.43
181.76
192.34
202.98
212.34
227.65
232.43
242.98
252.34
261.54
272.00
281.23
291.18
300.56

Figure 3 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 60 CNY

TermAmount of Money disbursed (in CNY)
13
23.23
32.34
42.65
53.87
63.45
72.12
81.54
93.76
103.98
113.56
121.23
130.45
140.34
151.54
162.34
171.23
183.45
193
201.32
211.54
223.65
233.87
242.56
252.54
2610.43
2710.54
282.34
291.65
302.48

Figure 4 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 90 CNY

TermAmount of Money disbursed (in CNY)
13.54
21.76
34.65
47.65
55.43
63.34
72.43
82.64
92.76
103.54
112.43
122.98
131.65
141.65
151.54
160.98
170.76
182.54
192.86
201.54
213.65
224.87
234.65
248.76
2515.43
262.54
2710.98
283.54
294.54
304.37

Figure 5 - Table On Raw Data Table For Disbursement Of Money In 30 Drawings When Total Amount Of Money Is 120 CNY

TermAmount of Money disbursed (in CNY)
15.03
25.02
35.76
44.36
55.03
65.32
75.65
84.87
95.34
105.76
114.67
125.09
135.87
144.56
154.56
164.87
175.67
185.23
195.67
205.98
214.87
224.67
234.56
244.98
255.87
265.89
275.03
284.45
294.34
301.03

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

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

1.230.8970.3330.1108890.12362207
1.430.8730.5570.3102490.35538259
0.971.071-0.1010.0102010.00952474
0.451.184-0.7340.5387560.45503041
0.781.127-0.3470.1204090.10684028
0.560.978-0.4180.1747240.1786544
1.230.8810.3490.1218010.13825312
4.510.9993.51112.32712112.3394605
0.980.9730.0074.9E-055.036E-05
0.231.137-0.9070.8226490.72352595
0.340.823-0.4830.2332890.28346173
0.210.677-0.4670.2180890.32214032
2.340.711.632.65693.74211268
0.980.5840.3960.1568160.26852055
10.6860.3140.0985960.14372595
0.650.751-0.1010.0102010.01358322
1.020.6740.3460.1197160.17762018
0.560.903-0.3430.1176490.13028682
0.980.99-0.010.00010.00010101
0.340.811-0.4710.2218410.27354007
4.231.1093.1219.7406418.7832651
0.341.412-1.0721.1491840.81386969
0.161.045-0.8850.7832250.74949761
0.541.321-0.7810.6099610.46174186
1.021.813-0.7930.6288490.34685549
0.941.423-0.4830.2332890.16394167
0.561.941-1.3811.9071610.9825662
0.450.801-0.3510.1232010.15380899
0.450.811-0.3610.1303210.16069174
0.520.597-0.0770.0059290.00993132
0.651.793-1.1431.3064490.72863859
1.651.745-0.0950.0090250.00517192
2.342.1410.1990.0396010.0184965
2.652.3680.2820.0795240.03358277
1.82.255-0.4550.2070250.0918071
21.9560.0440.0019360.00098978
1.781.7610.0190.0003610.000205
1.431.999-0.5690.3237610.16196148
1.761.947-0.1870.0349690.01796045
3.542.2731.2671.6052890.70624241
1.341.645-0.3050.0930250.05655015
0.651.355-0.7050.4970250.36680812
0.341.42-1.081.16640.82140845
1.231.1680.0620.0038440.0032911
1.651.3720.2780.0772840.05632945
2.431.5030.9270.8593290.57174251
1.431.3480.0820.0067240.00498813
1.761.805-0.0450.0020250.00112188
2.341.980.360.12960.06545455
2.981.6211.3591.8468811.1393467
2.342.2170.1230.0151290.00682409
7.652.8244.82623.2902768.24726487
2.432.0890.3410.1162810.05566348
2.982.6430.3370.1135690.04296973
2.343.627-1.2871.6563690.45667742
1.542.845-1.3051.7030250.59860281
23.881-1.8813.5381610.9116622
1.231.601-0.3710.1376410.08597189
1.181.621-0.4410.1944810.11997594
0.561.195-0.6350.4032250.33742678
32.690.310.09610.03572491
3.232.6180.6120.3745440.14306494
2.343.212-0.8720.7603840.23673225
2.653.552-0.9020.8136040.22905518
3.873.3820.4880.2381440.07041514
3.452.9340.5160.2662560.09074847
2.122.642-0.5220.2724840.1031355
1.542.998-1.4582.1257640.70906071
3.762.920.840.70560.24164384
3.983.410.570.32490.09527859
3.562.4681.0921.1924640.48317018
1.232.032-0.8020.6432040.3165374
0.452.13-1.682.82241.32507042
0.341.752-1.4121.9937441.13798174
1.542.058-0.5180.2683240.13038095
2.342.2540.0860.0073960.00328128
1.232.022-0.7920.6272640.31021958
3.452.7080.7420.5505640.20331019
32.970.030.00090.00030303
1.322.432-1.1121.2365440.50844737
1.543.326-1.7863.1897960.95904871
3.654.236-0.5860.3433960.0810661
3.873.1340.7360.5416960.17284493
2.563.964-1.4041.9712160.49727952
2.545.44-2.98.411.54595588
10.434.2686.16237.9702448.89649578
10.545.8224.71822.2595243.82334662
2.342.402-0.0620.0038440.00160033
1.652.432-0.7820.6115240.25144901
2.481.7920.6880.4733440.26414286
3.543.587-0.0470.0022090.00061583
1.763.491-1.7312.9963610.85831023
4.654.2830.3670.1346890.03144735
7.654.7362.9148.4913961.79294679
5.434.5090.9210.8482410.18812176
3.343.912-0.5720.3271840.08363599
2.433.523-1.0931.1946490.33909991
2.643.997-1.3571.8414490.46070778
2.763.893-1.1331.2836890.32974287
3.544.547-1.0071.0140490.22301495
2.433.291-0.8610.7413210.22525706
2.982.7090.2710.0734410.02711
1.652.84-1.191.41610.49862676
1.652.336-0.6860.4705960.20145377
1.542.744-1.2041.4496160.52828571
0.983.005-2.0254.1006251.36460067
0.762.696-1.9363.7480961.39024332
2.543.611-1.0711.1470410.3176519
2.863.96-1.11.210.30555556
1.543.243-1.7032.9002090.89429818
3.654.435-0.7850.6162250.13894589
4.875.648-0.7780.6052840.10716785
4.654.1790.4710.2218410.05308471
8.765.2853.47512.0756252.28488647
15.437.2538.17766.8633299.2187135
2.545.691-3.1519.9288011.74464962
10.987.7633.21710.3490891.3331301
3.543.2030.3370.1135690.03545707
4.543.2431.2971.6822090.51872001
4.372.3891.9813.9243611.64267936
5.034.4830.5470.2992090.06674303
5.024.3630.6570.4316490.09893399
5.765.3530.4070.1656490.03094508
4.365.92-1.562.43360.41108108
5.035.637-0.6070.3684490.0653626
5.324.890.430.18490.03781186
5.654.4031.2471.5550090.35317034
4.874.997-0.1270.0161290.00322774
5.344.8670.4730.2237290.04596856
5.765.6830.0770.0059290.00104329
4.674.1130.5570.3102490.07543132
5.093.3871.7032.9002090.85627665
5.873.552.325.38241.51616901
4.562.921.642.68960.92109589
4.563.431.131.27690.37227405
4.873.7571.1131.2387690.32972292
5.673.372.35.291.56973294
5.234.5130.7170.5140890.11391292
5.674.950.720.51840.10472727
5.984.0531.9273.7133290.9161927
4.875.543-0.6730.4529290.08171189
4.677.06-2.395.71210.80907932
4.565.223-0.6630.4395690.08416025
4.986.607-1.6272.6471290.40065521
5.879.067-3.19710.2208091.12725367
5.897.113-1.2231.4957290.21028103
5.039.703-4.67321.8369292.25053375
4.454.0030.4470.1998090.04991481
4.344.0530.2870.0823690.02032297
1.032.987-1.9573.8298491.28217241

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

TermAmount Drawn (CNY)Residual Drawing (CNY)Average Residual Amount (CNY)Maximum Share Possible (CNY)Probability
00301.0002.100.005
11.2328.770.9922.080.005
21.4327.340.9762.050.005
30.9726.370.9772.050.005
40.4525.920.9972.090.005
50.7825.141.0062.110.005
60.5624.581.0242.150.005
71.2323.351.0152.130.005
84.5118.840.8561.800.006
90.9817.860.8501.790.006
100.2317.630.8821.850.005
110.3417.290.9101.910.005
120.2117.080.9491.990.005
132.3414.740.8671.820.005
140.9813.760.8601.810.006
151.0012.760.8511.790.006
160.6512.110.8651.820.006
171.0211.090.8531.790.006
180.5610.530.8781.840.005
190.989.550.8681.820.005
200.349.210.9211.930.005
214.234.980.5531.160.009
220.344.640.5801.220.008
230.164.480.6401.340.007
240.543.940.6571.380.007
251.022.920.5841.230.008
260.941.980.4951.040.010
270.561.420.4730.990.010
280.450.970.4851.020.010
290.450.520.5201.090.009
300.52000.5200

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

For Graph 2

TermAmount Drawn (CNY)Residual Drawing (CNY)Average Residual Amount (CNY)Maximum Share Possible (CNY)Probability
00602.004.200.002
10.6559.352.054.300.002
21.6557.72.064.330.002
32.3455.362.054.310.002
42.6552.712.034.260.002
51.850.912.044.280.002
6248.912.044.280.002
71.7847.132.054.300.002
81.4345.72.084.360.002
91.7643.942.094.390.002
103.5440.42.024.240.002
111.3439.062.064.320.002
120.6538.412.134.480.002
130.3438.072.244.700.002
141.2336.842.304.840.002
151.6535.192.354.930.002
162.4332.762.344.910.002
171.4331.332.415.060.002
181.7629.572.465.170.002
192.3427.232.485.200.002
202.9824.252.435.090.002
212.3421.912.435.110.002
227.6514.261.783.740.003
232.4311.831.693.550.003
242.988.851.483.100.003
252.346.511.302.730.004
261.544.971.242.610.004
2722.970.992.080.005
281.231.740.871.830.005
291.180.560.561.180.009
300.56000.000

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

For Graph 3

TermAmount Drawn (CNY)Residual Drawing (CNY)Average Residual Amount (CNY)Maximum Share Possible (CNY)Probability
00903.0006.300.0016
13873.0006.300.0016
23.2383.772.9926.280.0016
32.3481.433.0166.330.0016
42.6578.783.0306.360.0016
53.8774.912.9966.290.0016
63.4571.462.9786.250.0016
72.1269.343.0156.330.0015
81.5467.83.0826.470.0016
93.7664.043.0506.400.0016
103.9860.063.0036.300.0016
113.5656.52.9746.240.0016
121.2355.273.0716.440.0015
130.4554.823.2256.770.0014
140.3454.483.4057.150.0013
151.5452.943.5297.410.0013
162.3450.63.6147.590.0013
171.2349.373.7987.970.0012
183.4545.923.8278.030.0012
19342.923.9028.190.0011
201.3241.64.1608.730.0011
211.5440.064.4519.340.0010
223.6536.414.5519.550.0010
233.8732.544.6499.760.0010
242.5629.984.99710.430.0009
252.5427.445.48811.550.0011
2610.4317.014.2528.900.0022
2710.546.472.1574.290.0023
282.344.132.0654.360.0019
291.652.482.4805.080.0016
302.48000.000.0016

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

For Graph 4:

TermAmount Drawn (CNY)Residual Drawing (CNY)Average Residual Amount (CNY)Maximum Share Possible (CNY)Probability
0012048.40.0012
13.54116.464.015862078.433310340.0012
21.76114.74.096428578.60250.0012
34.65110.054.075925938.559444440.0012
47.65102.43.938461548.270769230.0012
55.4396.973.87888.145480.0012
63.3493.633.901258.1926250.0012
72.4391.23.965217398.326956520.0012
82.6488.564.025454558.453454550.0012
92.7685.84.085714298.580.0012
103.5482.264.1138.63730.0012
112.4379.834.201578958.823315790.0011
122.9876.854.269444448.965833330.0011
131.6575.24.423529419.289411760.0011
141.6573.554.5968759.65343750.0010
151.5472.014.8006666710.08140.0010
160.9871.035.0735714310.65450.0009
170.7670.275.4053846211.35130770.0009
182.5467.735.6441666711.852750.0008
192.8664.875.8972727312.38427270.0008
201.5463.336.33313.29930.0008
213.6559.686.6311111113.92533330.0007
224.8754.816.8512514.3876250.0007
234.6550.167.1657142915.0480.0007
248.7641.46.914.490.0007
2515.4325.975.19410.90740.0009
262.5423.435.857512.300750.0008
2710.9812.454.158.7150.0011
283.548.914.4559.35550.0011
294.544.374.379.1770.0011
304.370000.0000

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

For Graph 5

TermAmount Drawn (CNY)Residual Drawing (CNY)Average Residual Amount (CNY)Maximum Share Possible (CNY)Probability
001505.00010.5000.00095
15.03144.974.99910.4980.00095
25.02139.954.99810.4960.00095
35.76134.194.97010.4370.00096
44.36129.834.99310.4860.00095
55.03124.84.99210.4830.00095
65.32119.484.97810.4550.00096
75.65113.834.94910.3930.00096
84.87108.964.95310.4010.00096
95.34103.624.93410.3620.00097
105.7697.864.89310.2750.00097
114.6793.194.90510.3000.00097
125.0988.14.89410.2780.00097
135.8782.234.83710.1580.00098
144.5677.674.85410.1940.00098
154.5673.114.87410.2350.00098
164.8768.244.87410.2360.00098
175.6762.574.81310.1070.00099
185.2357.344.77810.0350.00100
195.6751.674.6979.8640.00101
205.9845.694.5699.5950.00104
214.8740.824.5369.5250.00105
224.6736.154.5199.4890.00105
234.5631.594.5139.4770.00106
244.9826.614.4359.3130.00107
255.8720.744.1488.7110.00115
265.8914.853.7127.7960.00128
275.039.823.2736.8740.00145
284.455.372.6855.6380.00177
294.341.031.0302.1630.00462
301.0300.0000.0000.00000

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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