# Caveman Keno Max

## Introduction

Caveman Keno Max plays like Caveman Keno, except there are four eggs instead of three. For those not familiar with standard Caveman Keno, both games distribute eggs on numbers unchosen by the player. If the 20-number ball draw matches enough eggs, the player wins a multiplier.

## Rules

1. The game is played on four cards.
2. The player bets 5 to 40 credits, in increments of 5.
3. The player chooses 3 to 10 numbers per card, from the usual range of 1 to 80.
4. After the player bets, the game will choose FOUR numbers per card from among the numbers unchosen by the player. These numbers shall be marked with eggs.
5. The game shall draw 20 numbers, as usual, from a range of 1 to 80. This draw shall apply to all four player cards.
6. If the ball draw matches a number chosen by the player, it is known as a "catch."
7. If the ball draw matches a number with an egg, then that egg will hatch.
8. The player shall earn a multiplier according to the number of eggs that hatch on each card, as follows:
• 0 or 1 egg: 1x
• 2 eggs: 2x
• 3 eggs: 6x
• 4 eggs: 12x

The following pay table is used at VideoPoker.com and for a 5-credit bet. Please be advised this is likely the most liberal pay table available for the game. At the time of this writing, I do not know any other pay tables.

### VideoPoker.com Pay Table

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 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
2 3 1 1 0 0 0 0 0
3 31 6 2 2 1 1 1 0
4   130 15 6 5 3 1 1
5     390 68 24 8 3 6
6       2,000 270 70 38 18
7         2,000 700 285 110
8           2,000 1,500 500
9             2,000 1,500
10               2,000

## Example

In the image above, I bet five credits. Here is how I did on each card:

• Card A: I caught 2 out of 4, which normally pays 6. I also caught 2 out of 4 eggs, for a 2x multiplier. My total win for that card is 6 × 2 = 12.
• Card B: I caught 0 out of 6, which normally pays 0. I also caught 1 out of 4 eggs, for a 1x multiplier. My total win for that card is 0 × 1 = 0.
• Card C: I caught 4 out of 8, which normally pays 3. I also caught 0 out of 4 eggs, for a 1x multiplier. My total win for that card is 3 × 1 = 3.
• Card D: I caught 4 out of 10, which normally pays 1. I also caught 3 out of 4 eggs, for a 6x multiplier. My total win for that card is 1 × 6 = 6.

The image above is of card A. Note I caught 3 out of 4 and 2 out of 4 eggs hatched.

The image above is of card D. Note I caught 4 out of 10 and 3 out of 4 eggs hatched.

## Analysis

The following table shows the probability of any given combination of picks and catches, before considering the effect of the eggs.

### Probabilities in Keno

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 0.000000 0.003063 0.012092 0.028538 0.052191 0.081504 0.114105 0.147319
5 0.000000 0.000000 0.000645 0.003096 0.008639 0.018303 0.032601 0.051428
6 0.000000 0.000000 0.000000 0.000129 0.000732 0.002367 0.005720 0.011479
7 0.000000 0.000000 0.000000 0.000000 0.000024 0.000160 0.000592 0.001611
8 0.000000 0.000000 0.000000 0.000000 0.000000 0.000004 0.000033 0.000135
9 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000001 0.000006
10 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

After a lot of math, the following table shows the average multiplier, according to the number of picks and catches.

### Average Multiplier

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 1.503545 1.518401 1.533913 1.550122 1.567069 1.584799 1.603359 1.622803
1 1.448031 1.461207 1.474965 1.489339 1.504366 1.520086 1.536541 1.553778
2 1.396197 1.407809 1.419933 1.432599 1.445839 1.459689 1.474184 1.489367
3 1.347971 1.358133 1.368742 1.379825 1.391408 1.403523 1.416202 1.429480
4   1.312103 1.321313 1.330933 1.340987 1.351500 1.362502 1.374023
5     1.277564 1.285840 1.294488 1.303531 1.312993 1.322900
6       1.244459 1.251823 1.259522 1.267577 1.276009
7         1.212898 1.219377 1.226154 1.233247
8           1.182997 1.188621 1.194508
9             1.154873 1.159680
10               1.128652

The next table shows the product, for any combination of picks and catches, of the probability, average multiplier, and 0.8. Why multiply by 0.8? It is because for every five credits bet, one goes to each of the four cards and the other vanishes. In other words, 20% of the player's bet is like a fee for the egg feature.

If you find another pay table other than the one above, you may apply the factors below to get the return.

### Product of Probability, Average Multiplier, and 0.8.

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 0.500986 0.374524 0.278785 0.206602 0.152412 0.111907 0.081769 0.059447
1 0.499127 0.505849 0.478698 0.433093 0.379332 0.324039 0.271249 0.223211
2 0.154982 0.239480 0.307225 0.353361 0.377831 0.383192 0.373176 0.351797
3 0.014963 0.046989 0.091908 0.143303 0.194790 0.241166 0.278832 0.305797
4   0.003216 0.012782 0.030386 0.055990 0.088122 0.124375 0.161936
5     0.000659 0.003184 0.008946 0.019086 0.034244 0.054427
6       0.000128 0.000733 0.002385 0.005800 0.011718
7         0.000024 0.000157 0.000580 0.001590
8           0.000004 0.000031 0.000129
9             0.000001 0.000006
10               0.000000

The next table shows the contribution to the return for any combination of picks and catches. The bottom row shows the overall return. You can see it is highest at 94.77% for picking 10. Remember, this analysis is based on the VideoPoker.com pay table. Pay tables used on real casinos probably are not as generous.

### 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 0.464946 0.239480 0.307225 0.000000 0.000000 0.000000 0.000000 0.000000
3 0.463849 0.281935 0.183817 0.286605 0.194790 0.241166 0.278832 0.000000
4 0.000000 0.418026 0.191733 0.182314 0.279950 0.264365 0.124375 0.161936
5 0.000000 0.000000 0.257067 0.216539 0.214703 0.152691 0.102733 0.326562
6 0.000000 0.000000 0.000000 0.256826 0.197949 0.166932 0.220399 0.210928
7 0.000000 0.000000 0.000000 0.000000 0.047357 0.109567 0.165411 0.174850
8 0.000000 0.000000 0.000000 0.000000 0.000000 0.008225 0.046488 0.064704
9 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.001338 0.008518
10 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000203
Return 0.928795 0.939441 0.939842 0.942284 0.934748 0.942947 0.939578 0.947700