# Klub Keno

## Introduction

Internet casinos using Gamesys software offer a game called "Klub Keno." It is like 40-ball keno, except if the special "star ball" matches one of the player's picks, then the win is doubled.

## Rules

The following are my understanding of the rules. I couldn't find the rules anywhere in writing so I had to go off of observation. The "star ball" sometimes doesn't have a star on it, which made this difficult. If these rules are in error, then I welcome correction.

1. There are 40 balls, numbered 1 to 40.
2. The player shall pick 3 to 10 numbers.
3. One of the 40 balls shall be randomly chosen as the "Star Ball."
4. The game shall draw 10 numbers randomly.
5. The player shall be paid according to how many of the random draw match his chosen numbers.
6. If the Star Ball is drawn and matches one of the player's picks, then any win shall be doubled.

The following table shows what each number of catches pays by the number of picks.

### Pay TableExpand

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
3 4 5 6 7 8 9 10
0
1
2 3 2 1 1
3 40 4 3 2 1 1 1
4 130 20 4 5 5 2 1
5 450 80 50 8 5 5
6 1000 450 100 25 25
7 1300 800 700 450
8 8000 7000 2000
9 10000 8000
10 15000

## Analysis

The following table shows the number of combinations for each number of picks and catches.

### Combination TableExpand

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 4060 27405 142506 593775 2035800 5852925 14307150 30045015
1 4350 40600 274050 1425060 5937750 20358000 58529250 143071500
2 1350 19575 182700 1233225 6412770 26719875 91611000 263381625
3 120 3600 52200 487200 3288600 17100720 71253000 244296000
4 210 6300 91350 852600 5755050 29926260 124692750
5 252 7560 109620 1023120 6906060 35911512
6 210 6300 91350 852600 5755050
7 120 3600 52200 487200
8 45 1350 19575
9 10 300
10 1
Total 9880 91390 658008 3838380 18643560 76904685 273438880 847660528

The next table shows the probability, to six decimal points, for each number of picks and catches.

### Probability TableExpand

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 0.410931 0.299869 0.216572 0.154694 0.109196 0.076106 0.052323 0.035445
1 0.440283 0.44425 0.416484 0.371266 0.318488 0.264717 0.214049 0.168784
2 0.136640 0.214192 0.277656 0.321288 0.343967 0.347441 0.335033 0.310716
3 0.012146 0.039392 0.07933 0.126929 0.176393 0.222363 0.260581 0.288200
4 0.002298 0.009574 0.023799 0.045732 0.074834 0.109444 0.147102
5 0.000383 0.00197 0.00588 0.013304 0.025256 0.042365
6 0.000055 0.000338 0.001188 0.003118 0.006789
7 0.000006 0.000047 0.000191 0.000575
8 0.000001 0.000005 0.000023
9 0.000000 0.000000
10 0.000000
Total 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

The next table shows the contribution to the return, to six decimal points, for each number of picks and catches. This table reflects the probability and effect of catching the Star Ball. The bottom row shows the total return for each number of picks.

### Return TableExpand

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
2 0.430415 0.449803 0.291539 0.337352 0 0 0 0
3 0.522267 0.169384 0.25584 0.272896 0.189623 0.23904 0.280125 0
4 0.328592 0.210636 0.104716 0.251524 0.411584 0.240777 0.161812
5 0.193881 0.177262 0.330738 0.119734 0.142067 0.238306
6 0.062917 0.174873 0.136601 0.089644 0.195193
7 0.009832 0.044003 0.157017 0.303903
8 0.005617 0.041472 0.055423
9 0.000448 0.003468
10 0.000022
Total 0.952682 0.947779 0.951896 0.955144 0.956589 0.956579 0.951549 0.958129

## Strategy

As you can see in the table above, the greatest return is for the pick-10 at 95.81%.