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“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 |
---|---|---|---|---|---|---|---|---|---|---|

| --- | --- | 0.001 | 0.004 | 0.016 | 2.706 | 3.841 | 5.024 | 6.635 | 7.879 |

| 0.010 | 0.020 | 0.051 | 0.103 | 0.211 | 4.605 | 5.991 | 7.378 | 9.210 | 10.597 |

| 0.072 | 0.115 | 0.216 | 0.352 | 0.584 | 6.251 | 7.815 | 9.348 | 11.345 | 12.838 |

| 0.207 | 0.297 | 0.484 | 0.711 | 1.064 | 7.779 | 9.488 | 11.143 | 13.277 | 14.860 |

| 0.412 | 0.554 | 0.831 | 1.145 | 1.610 | 9.236 | 11.070 | 12.833 | 15.086 | 16.750 |

| 0.676 | 0.872 | 1.237 | 1.635 | 2.204 | 10.645 | 12.592 | 14.449 | 16.812 | 18.548 |

| 0.989 | 1.239 | 1.690 | 2.167 | 2.833 | 12.017 | 14.067 | 16.013 | 18.475 | 20.278 |

| 1.344 | 1.646 | 2.180 | 2.733 | 3.490 | 13.362 | 15.507 | 17.535 | 20.090 | 21.955 |

| 1.735 | 2.088 | 2.700 | 3.325 | 4.168 | 14.684 | 16.919 | 19.023 | 21.666 | 23.589 |

| 2.156 | 2.558 | 3.247 | 3.940 | 4.865 | 15.987 | 18.307 | 20.483 | 23.209 | 25.188 |

| 2.603 | 3.053 | 3.816 | 4.575 | 5.578 | 17.275 | 19.675 | 21.920 | 24.725 | 26.757 |

| 3.074 | 3.571 | 4.404 | 5.226 | 6.304 | 18.549 | 21.026 | 23.337 | 26.217 | 28.300 |

| 3.565 | 4.107 | 5.009 | 5.892 | 7.042 | 19.812 | 22.362 | 24.736 | 27.688 | 29.819 |

| 4.075 | 4.660 | 5.629 | 6.571 | 7.790 | 21.064 | 23.685 | 26.119 | 29.141 | 31.319 |

| 4.601 | 5.229 | 6.262 | 7.261 | 8.547 | 22.307 | 24.996 | 27.488 | 30.578 | 32.801 |

| 5.142 | 5.812 | 6.908 | 7.962 | 9.312 | 23.542 | 26.296 | 28.845 | 32.000 | 34.267 |

| 5.697 | 6.408 | 7.564 | 8.672 | 10.085 | 24.769 | 27.587 | 30.191 | 33.409 | 35.718 |

| 6.265 | 7.015 | 8.231 | 9.390 | 10.865 | 25.989 | 28.869 | 31.526 | 34.805 | 37.156 |

| 6.844 | 7.633 | 8.907 | 10.117 | 11.651 | 27.204 | 30.144 | 32.852 | 36.191 | 38.582 |

| 7.434 | 8.260 | 9.591 | 10.851 | 12.443 | 28.412 | 31.410 | 34.170 | 37.566 | 39.997 |

| 8.034 | 8.897 | 10.283 | 11.591 | 13.240 | 29.615 | 32.671 | 35.479 | 38.932 | 41.401 |

| 8.643 | 9.542 | 10.982 | 12.338 | 14.041 | 30.813 | 33.924 | 36.781 | 40.289 | 42.796 |

| 9.260 | 10.196 | 11.689 | 13.091 | 14.848 | 32.007 | 35.172 | 38.076 | 41.638 | 44.181 |

| 9.886 | 10.856 | 12.401 | 13.848 | 15.659 | 33.196 | 36.415 | 39.364 | 42.980 | 45.559 |

| 10.520 | 11.524 | 13.120 | 14.611 | 16.473 | 34.382 | 37.652 | 40.646 | 44.314 | 46.928 |

| 11.160 | 12.198 | 13.844 | 15.379 | 17.292 | 35.563 | 38.885 | 41.923 | 45.642 | 48.290 |

| 11.808 | 12.879 | 14.573 | 16.151 | 18.114 | 36.741 | 40.113 | 43.195 | 46.963 | 49.645 |

| 12.461 | 13.565 | 15.308 | 16.928 | 18.939 | 37.916 | 41.337 | 44.461 | 48.278 | 50.993 |

| 13.121 | 14.256 | 16.047 | 17.708 | 19.768 | 39.087 | 42.557 | 45.722 | 49.588 | 52.336 |

| 13.787 | 14.953 | 16.791 | 18.493 | 20.599 | 40.256 | 43.773 | 46.979 | 50.892 | 53.672 |

| 20.707 | 22.164 | 24.433 | 26.509 | 29.051 | 51.805 | 55.758 | 59.342 | 63.691 | 66.766 |

| 27.991 | 29.707 | 32.357 | 34.764 | 37.689 | 63.167 | 67.505 | 71.420 | 76.154 | 79.490 |

| 35.534 | 37.485 | 40.482 | 43.188 | 46.459 | 74.397 | 79.082 | 83.298 | 88.379 | 91.952 |

| 43.275 | 45.442 | 48.758 | 51.739 | 55.329 | 85.527 | 90.531 | 95.023 | 100.425 | 104.215 |

| 51.172 | 53.540 | 57.153 | 60.391 | 64.278 | 96.578 | 101.879 | 106.629 | 112.329 | 116.321 |

| 59.196 | 61.754 | 65.647 | 69.126 | 73.291 | 107.565 | 113.145 | 118.136 | 124.116 | 128.299 |

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

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 X^{2}

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