# Captain Coins Keno

## Introduction

Captain Coins Keno is a simple keno variant that awards a bonus round if the 20th ball drawn is part of a win. The bonus round consists of 12 free games with a 2x multiplier.

## Rules

1. The player shall pick 3 to 10 numbers from 1 to 80.
2. The game shall draw 20 numbers, without replacement, from 1 to 80.
3. If a ball drawn by the game matches a number chosen by the player, then it shall count as a "catch."
4. The player shall in according to the number of picks, catches, and the pay table below.
5. In addition, if the 20th ball drawn by the game, denoted with a parrot, the player shall be awarded a bonus round.
6. The bonus round shall consist of 12 games with a 2x multiplier.
7. The bet and number of picks are the same in the bonus round as the initiating game.
8. The bonus can not be re-triggered in a bonus. In other words, there is nothing special about the 20th ball in bonus games.

### Captain Coins Keno Pay Table

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 3 1 0 0 0 0 0 0
3 17 5 3 2 1 0 0 0
4   35 22 5 3 3 2 1
5     225 34 6 10 5 3
6       375 95 56 10 5
7         500 180 80 25
8           1000 200 150
9             1000 1000
10               2000

## Analysis

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

### Keno Combinations

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 34220 487635 5461512 50063860 386206920 2558620845 14783142660 75394027566
1 35400 684400 9752700 109230240 1001277200 7724138400 51172416900 295662853200
2 11400 336300 6501800 92650650 1037687280 9512133400 73379314800 486137960550
3 1140 68400 2017800 39010800 555903900 6226123680 57072800400 440275888800
4   4845 290700 8575650 165795900 2362591575 26461025640 242559401700
5     15504 930240 27442080 530546880 7560293040 84675282048
6       38760 2325600 68605200 1326367200 18900732600
7         77520 4651200 137210400 2652734400
8           125970 7558200 222966900
9             167960 10077600
10               184756
Total 82160 1581580 24040016 300500200 3176716400 28987537150 231900297200 1646492110120

The following table shows the probability for all possible combinations of picks and catches.

### Keno Probabilities

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 0.416504 0.308321 0.227184 0.166602 0.121574 0.088266 0.063748 0.045791
1 0.430867 0.432732 0.405686 0.363495 0.315193 0.266464 0.220666 0.179571
2 0.138754 0.212635 0.270457 0.308321 0.326654 0.328146 0.316426 0.295257
3 0.013875 0.043248 0.083935 0.129820 0.174993 0.214786 0.246109 0.267402
4   0.003063 0.012092 0.028538 0.052191 0.081504 0.114105 0.147319
5     0.000645 0.003096 0.008639 0.018303 0.032601 0.051428
6       0.000129 0.000732 0.002367 0.005720 0.011479
7         0.000024 0.000160 0.000592 0.001611
8           0.000004 0.000033 0.000135
9             0.000001 0.000006
10               0.000000
Total 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

The following table shows the contribution to the return for all possible combinations of picks and catches. The bottom row shows the overall return, not including the value of the bonus.

### Keno Return

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
2 0.416261 0.212635 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
3 0.235881 0.216239 0.251805 0.259639 0.174993 0.000000 0.000000 0.000000
4   0.107219 0.266031 0.142690 0.156573 0.244511 0.228210 0.147319
5     0.145108 0.105252 0.051831 0.183026 0.163007 0.154283
6       0.048369 0.069547 0.132536 0.057196 0.057397
7         0.012201 0.028882 0.047334 0.040279
8           0.004346 0.006518 0.020313
9             0.000724 0.006121
10               0.000224
Total 0.652142 0.536094 0.662945 0.555950 0.465146 0.593301 0.502990 0.425935

The following table shows the probability of triggering the bonus, the average bonus win, and contribution to the return from the bonus per initial bet.

### Bonus Summary

Pick Bonus
Probability
Average
Bonus Win
Bonus
Return
3 0.015957 15.651412 0.249744
4 0.028363 12.866248 0.364931
5 0.015170 15.910672 0.241364
6 0.025993 13.342794 0.346821
7 0.039075 11.163498 0.436213
8 0.021644 14.239212 0.308198
9 0.032908 12.071769 0.397254
10 0.046385 10.222452 0.474172

The following table shows how much the base game and bonus contribution to the return as well as the total. The right column shows the greatest return is for the pick 5 at 90.43%.

### Caption Coins Keno Summary

Pick Base Game
Return
Bonus
Return
Total
Return
3 0.652142 0.249744 0.901887
4 0.536094 0.364931 0.901024
5 0.662945 0.241364 0.904309
6 0.555950 0.346821 0.902771
7 0.465146 0.436213 0.901359
8 0.593301 0.308198 0.901498
9 0.502990 0.397254 0.900245
10 0.425935 0.474172 0.900108

## Other Pay Tables

There probably are other pay tables out there. The one seen above is the only one I have seen, which I saw at the Wynn Las Vegas on July 11, 2021. If I see other pay tables, I will update this page with that information.