# Top, Bottom, Left, Right and Edge Keno

## Introduction

Top Bottom Keno is a keno game in which the player selects either numbers 1 to 40 or 41 to 80. He is paid according to the number of catches in the chosen half.  Sometimes the player can also bet on the left side, right side, and edge numbers, which are also addressed.

## Rules

1. The player shall pick either Top or Bottom. If the player picks Top, then he shall cover numbers 1 to 40. If he picks Bottom, then 41 to 80.
2. The game shall draw 20 balls randomly and without replacement from a pool of balls numbered 1 to 80.
3. The player shall be paid according to how many balls in the 20-ball draw that fall in the player's chosen half. This is known as the number of catches. An example pay table is below.

The following table shows the three Pay Tables (PT) I am aware of. The bottom row shows the return to player (RTP).

### Pay Tables

Catches PT 1 PT 2 PT 3 PT 4
0 or 20 12500 12500 12500 12500
1 or 19 5000 5000 5000 5000
2 or 18 492 434 558 1000
3 or 17 122 125 180 200
4 or 16 28 30 32 40
5 or 15 11 12 12 10
6 or 14 3 3 3 3
7 or 13 2 2 2 1
12 1 1 1 0
Return 88.32% 90.32% 94.32% 72.29%

## Example

In the image below I bet the bottom 40 numbers. I bet one credit and caught 16 out of 20, for a win of 32.

## Analysis

The following table shows the probability and contribution to the return of all possible number of catches from 0 to 20 for pay table 1. The lower right cell shows a return of 88.32%.

### Pay Table 1 Analysis

Catches Pays Combinations Probability Return
0 12,500 137,846,528,820 0.00000004 0.000487
1 5,000 5,251,296,336,000 0.00000149 0.007427
2 492 88,436,604,204,000 0.00002502 0.012307
3 122 876,675,902,544,001 0.00024798 0.030253
4 28 5,744,053,569,793,500 0.00162476 0.045493
5 11 26,468,598,849,608,400 0.00748691 0.082356
6 3 89,077,015,359,259,200 0.02519634 0.075589
7 2 224,342,112,756,653,000 0.06345744 0.126915
8 - 429,655,207,020,554,000 0.12153233 0.000000
9 - 632,136,396,535,987,000 0.17880619 0.000000
10 - 718,528,370,729,238,000 0.20324303 0.000000
11 - 632,136,396,535,987,000 0.17880619 0.000000
12 1 429,655,207,020,554,000 0.12153233 0.121532
13 2 224,342,112,756,653,000 0.06345744 0.126915
14 3 89,077,015,359,259,200 0.02519634 0.075589
15 11 26,468,598,849,608,400 0.00748691 0.082356
16 28 5,744,053,569,793,500 0.00162476 0.045493
17 122 876,675,902,544,001 0.00024798 0.030253
18 492 88,436,604,204,000 0.00002502 0.012307
19 5,000 5,251,296,336,000 0.00000149 0.007427
20 12,500 137,846,528,820 0.00000004 0.000487
Total   3,535,316,142,212,170,000 1.00000000 0.883189

The following table shows the probability and contribution to the return of all possible number of catches from 0 to 20 for pay table 1. The lower right cell shows a return of 90.32%.

### Pay Table 2 Analysis

Catches Pays Combinations Probability Return
0 12,500 137,846,528,820 0.000000 0.000487
1 5,000 5,251,296,336,000 0.000001 0.007427
2 434 88,436,604,204,000 0.000025 0.010857
3 125 876,675,902,544,001 0.000248 0.030997
4 30 5,744,053,569,793,500 0.001625 0.048743
5 12 26,468,598,849,608,400 0.007487 0.089843
6 3 89,077,015,359,259,200 0.025196 0.075589
7 2 224,342,112,756,653,000 0.063457 0.126915
8 - 429,655,207,020,554,000 0.121532 0.000000
9 - 632,136,396,535,987,000 0.178806 0.000000
10 - 718,528,370,729,238,000 0.203243 0.000000
11 - 632,136,396,535,987,000 0.178806 0.000000
12 1 429,655,207,020,554,000 0.121532 0.121532
13 2 224,342,112,756,653,000 0.063457 0.126915
14 3 89,077,015,359,259,200 0.025196 0.075589
15 12 26,468,598,849,608,400 0.007487 0.089843
16 30 5,744,053,569,793,500 0.001625 0.048743
17 125 876,675,902,544,001 0.000248 0.030997
18 434 88,436,604,204,000 0.000025 0.010857
19 5,000 5,251,296,336,000 0.000001 0.007427
20 12,500 137,846,528,820 0.000000 0.000487
Total   3,535,316,142,212,170,000 1.000000 0.903248

The following table shows the probability and contribution to the return of all possible number of catches from 0 to 20 for pay table 3. The lower right cell shows a return of 94.32%.

### Pay Table 3 Analysis

Catches Pays Combinations Probability Return
0 12,500 137,846,528,820 0.000000 0.000487
1 5,000 5,251,296,336,000 0.000001 0.007427
2 558 88,436,604,204,000 0.000025 0.013958
3 180 876,675,902,544,001 0.000248 0.044636
4 32 5,744,053,569,793,500 0.001625 0.051992
5 12 26,468,598,849,608,400 0.007487 0.089843
6 3 89,077,015,359,259,200 0.025196 0.075589
7 2 224,342,112,756,653,000 0.063457 0.126915
8 - 429,655,207,020,554,000 0.121532 0.000000
9 - 632,136,396,535,987,000 0.178806 0.000000
10 - 718,528,370,729,238,000 0.203243 0.000000
11 - 632,136,396,535,987,000 0.178806 0.000000
12 1 429,655,207,020,554,000 0.121532 0.121532
13 2 224,342,112,756,653,000 0.063457 0.126915
14 3 89,077,015,359,259,200 0.025196 0.075589
15 12 26,468,598,849,608,400 0.007487 0.089843
16 32 5,744,053,569,793,500 0.001625 0.051992
17 180 876,675,902,544,001 0.000248 0.044636
18 558 88,436,604,204,000 0.000025 0.013958
19 5,000 5,251,296,336,000 0.000001 0.007427
20 12,500 137,846,528,820 0.000000 0.000487
Total   3,535,316,142,212,170,000 1.000000 0.943228

The following table shows the probability and contribution to the return of all possible number of catches from 0 to 20 for pay table 4. The lower right cell shows a return of 72.29%.

### Pay Table 4 Analysis

Catches Pays Combinations Probability Return
0 12,500 137,846,528,820 0.000000 0.000487
1 5,000 5,251,296,336,000 0.000001 0.007427
2 1,000 88,436,604,204,000 0.000025 0.025015
3 200 876,675,902,544,001 0.000248 0.049595
4 40 5,744,053,569,793,500 0.001625 0.064991
5 10 26,468,598,849,608,400 0.007487 0.074869
6 3 89,077,015,359,259,200 0.025196 0.075589
7 1 224,342,112,756,653,000 0.063457 0.063457
8 - 429,655,207,020,554,000 0.121532 0.000000
9 - 632,136,396,535,987,000 0.178806 0.000000
10 - 718,528,370,729,238,000 0.203243 0.000000
11 - 632,136,396,535,987,000 0.178806 0.000000
12 - 429,655,207,020,554,000 0.121532 0.000000
13 1 224,342,112,756,653,000 0.063457 0.063457
14 3 89,077,015,359,259,200 0.025196 0.075589
15 10 26,468,598,849,608,400 0.007487 0.074869
16 40 5,744,053,569,793,500 0.001625 0.064991
17 200 876,675,902,544,001 0.000248 0.049595
18 1,000 88,436,604,204,000 0.000025 0.025015
19 5,000 5,251,296,336,000 0.000001 0.007427
20 12,500 137,846,528,820 0.000000 0.000487
Total   3,535,316,142,212,170,000 1.000000 0.722862

## Edge

I saw a bet on the edge at the Plaza in Las Vegas on June 13, 2023. The ticket price was \$3. The following table shows what each number of catches along the edge pay (based on a \$3 bet), the probability, and return. The return is based on a \$3 bet. The lower right cell shows an expected return of 71.38%.

### Edge Analysis

Catches Pays (\$3 bet) Combinations Probability Return
0 15000 16,735,679,449,896 0.000005 0.023669
1 1800 369,339,132,687,360 0.000104 0.062683
2 180 3,625,679,152,547,580 0.001026 0.061534
3 18 21,052,330,563,179,500 0.005955 0.035729
4 6 81,084,366,934,746,100 0.022936 0.045871
5 3 220,156,341,737,977,000 0.062273 0.062273
6 0 437,075,090,215,102,000 0.123631 0.000000
7 0 649,368,705,462,438,000 0.183681 0.000000
8 0 732,794,546,094,764,000 0.207278 0.000000
9 0 633,768,256,081,958,000 0.179268 0.000000
10 0 421,956,233,654,567,000 0.119355 0.000000
11 3 216,387,812,130,547,000 0.061207 0.061207
12 9 85,202,701,026,403,000 0.024100 0.072301
13 30 25,576,795,805,299,200 0.007235 0.072347
14 120 5,785,227,622,627,200 0.001636 0.065456
15 600 968,689,276,346,880 0.000274 0.054801
16 6000 116,958,222,286,200 0.000033 0.066166
17 30000 9,784,740,165,120 0.000003 0.027677
18 40000 531,779,356,800 0.000000 0.002006
19 75000 16,673,932,800 0.000000 0.000118
20 100000 225,792,840 0.000000 0.000002
Total   3,535,316,142,212,170,000 1.000000 0.713840