Lottery

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    To each lottery we can attach a set of basic numerical parameters through which that lottery is uniquely identified. These parameters are as follows:
- the total number of the numbers from the urn, further denoted by m;
- the number of numbers of a draw, further denoted by n;
- the number of numbers on a simple playing ticket (or the number of numbers of a simple line), further denoted by p;
- the number of winning categories, further denoted by q;
- the minimum number of winning numbers for each winning category, further denoted by
 (in decreasing order of the amount of the prize);
- the price of a simple ticket (simple line), further denoted by c;
- the percentage from sales for the prize fund, further denoted by f;
- the percentages in which the prize fund is distributed to each winning category, further denoted by
.
Any numerical instance of the vector (m, n, p, q,
, c, f, ) will be called a lottery matrix.

General formula of the probability of winning  with a simple line

Consider a played simple line. Let  and let  be the event  “exactly w numbers from the p given numbers are drawn.” The probability is given by the formula:

Below are the numerical returns of the above formula for n = p = 6,  and m from 21 up to 55.

n = 6

 w

m

0

1

2

3

4

5

6

21

0.092234

0.332043

0.377322

0.167699

0.029025

0.001659

1.84E-05

22

0.107327

0.351252

0.365888

0.150108

0.024124

0.001287

1.34E-05

23

0.122599

0.367797

0.353651

0.134724

0.020209

0.00101

9.91E-06

24

0.137924

0.381943

0.341021

0.121252

0.017051

0.000802

7.43E-06

25

0.153202

0.393947

0.328289

0.10943

0.014483

0.000644

5.65E-06

26

0.168353

0.404048

0.315663

0.099031

0.012379

0.000521

4.34E-06

27

0.183318

0.412466

0.303284

0.089862

0.010642

0.000426

3.38E-06

28

0.198049

0.419398

0.291249

0.081754

0.009197

0.00035

2.65E-06

29

0.212511

0.425022

0.27962

0.074565

0.007989

0.000291

2.11E-06

30

0.226678

0.429496

0.268435

0.068174

0.006972

0.000243

1.68E-06

31

0.240533

0.43296

0.257714

0.062476

0.006112

0.000204

1.36E-06

32

0.254063

0.435537

0.247464

0.057383

0.00538

0.000172

1.1E-06

33

0.267261

0.437337

0.237683

0.052818

0.004754

0.000146

9.03E-07

34

0.280124

0.438455

0.228362

0.048717

0.004216

0.000125

7.44E-07

35

0.292651

0.438977

0.219489

0.045023

0.003752

0.000107

6.16E-07

36

0.304845

0.438977

0.211047

0.041688

0.00335

9.24E-05

5.13E-07

37

0.316709

0.438521

0.203019

0.03867

0.003

8E-05

4.3E-07

38

0.328249

0.437666

0.195387

0.035933

0.002695

6.95E-05

3.62E-07

39

0.339472

0.436464

0.188131

0.033445

0.002427

6.07E-05

3.07E-07

40

0.350383

0.434958

0.181233

0.03118

0.002192

5.31E-05

2.61E-07

41

0.360992

0.43319

0.174674

0.029112

0.001985

4.67E-05

2.22E-07

42

0.371306

0.431194

0.168435

0.027222

0.001801

4.12E-05

1.91E-07

43

0.381334

0.429001

0.1625

0.02549

0.001639

3.64E-05

1.64E-07

44

0.391084

0.426637

0.156852

0.023901

0.001494

3.23E-05

1.42E-07

45

0.400565

0.424127

0.151474

0.022441

0.001365

2.87E-05

1.23E-07

46

0.409785

0.421493

0.146352

0.021096

0.001249

2.56E-05

1.07E-07

47

0.418753

0.418753

0.141471

0.019856

0.001146

2.29E-05

9.31E-08

48

0.427477

0.415923

0.136817

0.01871

0.001052

2.05E-05

8.15E-08

49

0.435965

0.413019

0.132378

0.01765

0.000969

1.84E-05

7.15E-08

50

0.444225

0.410054

0.128142

0.016669

0.000893

1.66E-05

6.29E-08

51

0.452266

0.407039

0.124097

0.015758

0.000825

1.5E-05

5.55E-08

52

0.460093

0.403984

0.120233

0.014913

0.000763

1.36E-05

4.91E-08

53

0.467716

0.400899

0.11654

0.014126

0.000706

1.23E-05

4.36E-08

54

0.47514

0.397791

0.113009

0.013394

0.000655

1.12E-05

3.87E-08

55

0.482372

0.394668

0.10963

0.012711

0.000608

1.01E-05

3.45E-08

 

Probabilities of winning with systems

A sufficient condition for the probability of winning (k winning numbers) with a system of several lines  to grow proportionally with the number of the constituent simple lines is that any two lines of the system will not contain more than 2kn – 1 common numbers. Generally, the winning probability has to be calculated by using the inclusion-exclusion principle. For systems under this sufficient condition, players may establish their own probability thresholds for the played systems. The chosen probability threshold () will provide through a simple calculation the minimal number of simple lines required for the system to offer the minimal probability chosen as threshold. But this number must be also limited by ensuring the profitability of the game (the eventual prize to be higher than the investment in the playing tickets). Next we present a table with minimal values of number of lines for reaching various winning probability thresholds for systems under the sufficient condition, for individual or cumulated winning categories, for the 6/49 lottery matrix.

6 /49 

k

4

5

6

min.4

min.5

1/100

11

544

139861

11

541

1/50

21

1087

279721

21

1082

1/30

35

1812

466201

34

1802

1/20

52

2718

699301

51

2703

1/10

104

5435

1398602

102

5406

1/8

129

6794

1748252

127

6757

1/7

148

7764

1998002

145

7723

1/6

172

9058

2331003

169

9010

1/5

207

10870

2797203

203

10811

1/4

258

13587

3496504

254

13514

1/3

344

18116

4662005

338

18019

 

Compound Lines

     A compound (combined) line is a playing system consisting of all possible simple lines that can be formed with a given number of playing numbers. Using our general denotations, a compound line is in fact a combination of r numbers, where .

Number r (the number of chosen playing numbers) is called the size of the compound line.  The playing system corresponding to a r-size compound line will cost the equivalent of all constituent simple lines together, namely  (c being the price of a simple line).

     Probability of winning

For evaluating the probability of occurrence in the draw of a certain number of winning numbers in the case of the play with one compound line, let us observe that the event “we will have exactly w winning numbers in at least one simple line of the system” is identical with “we will have exactly w winning numbers among the r of the initial combination,” where . This identity of events holds true because the simple lines of the system represent all possible p-size combinations of numbers from the played r. Denoting this event by , the probability  is given by the same general formula of the winning probability deduced in the section titled General formula of the probability of winning with a simple line, in which parameter r will replace parameter p:  

Next, we present a table containing the winning probabilities (including cumulated) for the play with one compound line, for the 6/49 lottery. In the tables in the first row are noted the values of the number of winning numbers (w) the probability refers to (including cumulated), and in the first column are noted the values of the size of the compound line (r). At the intersection of the column corresponding to a value of w with the row corresponding to a value of r is the probability of occurrence in the draw of w numbers from the played r. The second column holds the number of simple lines of the unfolded compound line, for every value of r.

6/49 

w

 

r

 

4

5

6

min.4

min.5

7

7

0.002155

6.31E-05

5.01E-07

0.002219

6.36E-05

8

28

0.004105

0.000164

2E-06

0.004271

0.000166

9

84

0.007028

0.00036

6.01E-06

0.007394

0.000366

10

210

0.011128

0.000703

1.5E-05

0.011846

0.000718

11

462

0.01659

0.001255

3.3E-05

0.017878

0.001288

12

924

0.023575

0.002096

6.61E-05

0.025737

0.002162

13

1716

0.032212

0.003313

0.000123

0.035648

0.003436

14

3003

0.042592

0.005011

0.000215

0.047818

0.005226

15

5005

0.054761

0.007301

0.000358

0.06242

0.007659

16

8008

0.068719

0.010308

0.000573

0.0796

0.010881

17

12376

0.084418

0.01416

0.000885

0.099463

0.015045

18

18564

0.101753

0.018994

0.001328

0.122074

0.020321

19

27132

0.120572

0.024946

0.00194

0.147458

0.026886

20

38760

0.140668

0.032153

0.002772

0.175592

0.034924

21

54264

0.161782

0.040745

0.00388

0.206408

0.044626

 

The number of prizes

A compound line may ensure – in case of winning in an arbitrary category – the simultaneous existence of several winnings in lower categories. The number of these winnings are given by the following formula:
If in an r-size compound line there are k winning numbers, then there exist  simple lines containing exactly ki numbers from the winning k, if .

 Next we present a table of values of number , returned by the general formula, for the lottery matrices with p = 5 and for a large enough range of values for the size of the compound line.

In this table, in the first row are noted the values of the number of winning numbers (k), in the second row are values of the number of winning numbers that are less than or equal to k (namely ki), and in the first column are the values of the size of the compound line (r). At the intersection of the row corresponding to a value of r with the column corresponding to a pair (k, ki) we find the number of simple lines containing exactly ki numbers from the winning k.

p = 5 

        k

r    k-i   

3

4

5

3

2

4

3

5

4

3

6

3

3

2

4

1

5

0

7

6

12

3

12

1

10

10

8

10

30

4

24

1

15

30

9

15

60

5

40

1

20

60

10

21

105

6

60

1

25

100

11

28

168

7

84

1

30

150

12

36

252

8

112

1

35

210

13

45

360

9

144

1

40

280

14

55

495

10

180

1

45

360

15

66

660

11

220

1

50

450

16

78

858

12

264

1

55

550

17

91

1092

13

312

1

60

660

18

105

1365

14

364

1

65

780

19

120

1680

15

420

1

70

910

20

136

2040

16

480

1

75

1050

21

153

2448

17

544

1

80

1200

22

171

2907

18

612

1

85

1360

23

190

3420

19

684

1

90

1530

 

Example of how to use the table:

In a 5/56 matrix, we played a 9-number compound line. What is the number of simple lines containing exactly 4 winning numbers, if exactly 5 winning numbers from the played 9 occurred in the draw? We follow the intersection of row r = 9 with the column corresponding to (k = 5; ki = 4) and find 20 simple lines.

       

 Sources

All tables for the most popular lotteries and the complete formulas of calculus, along with examples, are in the book THE MATHEMATICS OF LOTTERY: Odds, Combinations, Systems. They can be applied to any existent lottery and any playing system. The book discusses all playing systems, including the reduced systems. See the Books section for details.

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