Last Updated: September 1, 2016

# Top Bottom 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.

## 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 pay table was seen somewhere on the Las Vegas Strip (I don't remember the exact casino). All pays are on a "for one" basis, meaning the player does not also get back his original bet on a win.

Catches Pays
0 or 20 12,500
1 or 19 5,000
2 or 18 492
3 or 17 122
4 or 16 28
5 or 15 11
6 or 14 3
7 or 13 2
12 1

## Analysis

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

### 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

Written by: