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Last Updated: January 3, 2023

# Dream Ball Keno

## Introduction

Dream Ball Keno plays like standard keno, except sometimes the player gets an extra pick daubed on his card.

## Rules

1. The game is based on standard 4-card spot keno, which I assume the reader is familiar with.
2. The player may bet 2 to 10 coins per card, for a total bet of 8 to 40.
3. Randomly, the game will award the player an extra pick daubed on each card, unless every pick was already daubed. The following is the probability, according to the pay tables bellow:
• Pay Table 95 -- 10.6%
• Pay Table 93 -- 10.6%
• Pay Table 91 -- 10.5%
• Pay Table 90 -- 10.2%
• Pay Table 88 -- 10.0%

## Example

In the image above I bet 10 credits per card. The feature was enabled. Here is how many catches I had on each card, before the feature:

• Card A (Purple): 1 out of 4, win = 0
• Card B (Green): 2 out of 6, win = 0
• Card C (Yellow): 1 out of 8, win = 0
• Card D (Pink): 1 out of 10, win = 0

After the extra daub on each card, here is how many catches I had on each card:

• Card A (Purple): 2 out of 4, win = 10
• Card B (Green): 4 out of 6, win = 15
• Card C (Yellow): 2 out of 8, win = 0
• Card D (Pink): 2 out of 10, win = 0

So, the feature turned a "win" of 0 to one of 25.

## Analysis

The following table shows the probability of each number of catches, according to the number of picks, assuming the feature is not enabled.

### 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 what Pay Table 95 below, for each combination of picks and catches, assuming the feature is not enabled.

### Return Table without Feature

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 0.138754 0.212635 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
3 0.381573 0.172992 0.125903 0.194729 0.087497 0.000000 0.000000 0.000000
4   0.188399 0.078600 0.128421 0.078286 0.081504 0.057053 0.073659
5     0.224111 0.126921 0.185728 0.183026 0.179308 0.154283
6       0.090289 0.161789 0.130169 0.165867 0.200889
7         0.030503 0.144410 0.128690 0.138558
8           0.005432 0.054755 0.067710
9             0.002173 0.010711
10               0.000337
Total 0.520326 0.574026 0.428614 0.540361 0.543803 0.544541 0.587846 0.646148

The next table shows a expected under Pay Table 95, when the feature is enabled according to the number of picks and catches BEFORE the extra daub.

### Pay Table with Feature

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 0.430867 0.432732 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
2 3.815725 0.850542 0.405686 0.462482 0.163327 0.000000 0.000000 0.000000
3 0.381573 2.659745 0.545578 0.584188 0.262490 0.214786 0.123055 0.133701
4   0.188399 4.202087 1.170055 1.122106 0.815037 0.627579 0.441957
5     0.224111 2.166947 1.909110 1.006642 0.945443 0.899985
6       0.090289 0.915096 2.130042 1.244004 0.987228
7         0.030503 0.200569 0.994020 0.805572
8           0.005432 0.097777 0.236984
9             0.002173 0.018362
10               0.000337
Total 4.628165 4.131418 5.377463 4.473961 4.402631 4.372509 4.034050 3.524124

The final table shows the expected return both with and without the feature enabled, under Pay Table 95. Recall that the probability of the feature is 1 in 9.43 =~ 10.6%. The combined column is a weighted average between 89.4% of the time when the feature is not enabled and 10.6% when it is. You can see the overall return, under this pay table, is 95.05% to 95.59%.

### Pay Table with Feature

Pick Without
Feature
With
Feature
Weighted
Average
3 0.520326 4.628165 0.955940
4 0.574026 4.131418 0.951268
5 0.428614 5.377463 0.953412
6 0.540361 4.473961 0.957497
7 0.543803 4.402631 0.953011
8 0.544541 4.372509 0.950476
9 0.587846 4.034050 0.953297
10 0.646148 3.524124 0.951341

## Return Tables

The following pay tables are based on eight credits bet.

The following table is for what I call Pay Table 95. The feature probability is 10.6%. The bottom row shows the return for all number of picks. The greatest return is 95.73% for the pick 6.

### Pay Table 95

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 2 2 0 0 0 0 0 0
3 55 8 3 3 1 0 0 0
4 123 13 9 3 2 1 1
5 695 82 43 20 11 6
6 1400 442 110 58 35
7 2500 1800 435 172
8 2500 3360 1000
9 6000 3500
10 6000
Return 0.955757 0.951109 0.953192 0.957322 0.952839 0.950305 0.953144 0.951213

The following table is for what I call Pay Table 93. The feature probability is 10.6%. The bottom row shows the return for all number of picks. The greatest return is 93.96% for the pick 4.

### Pay Table 93

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 2 2 0 0 0 0 0 0
3 53 8 3 3 1 0 0 0
4 120 12 8 3 2 1 1
5 690 82 40 19 10 6
6 1400 442 110 58 34
7 2500 1800 435 166
8 2500 3360 1000
9 6000 3500
10 6000
Return 0.927174 0.939638 0.938521 0.937685 0.932956 0.937804 0.932523 0.935385

The following table is for what I call Pay Table 91. The feature probability is 10.5%. The bottom row shows the return for all number of picks. The greatest return is 91.64% for the pick 10.

### Pay Table 91

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 2 2 0 0 0 0 0 0
3 52 8 3 3 1 0 0 0
4 114 12 7 3 2 1 1
5 670 82 38 18 10 6
6 1405 442 107 56 33
7 2500 1800 430 160
8 2500 3360 1000
9 5000 3500
10 5000
Return 0.908982 0.913332 0.914493 0.915351 0.915951 0.915479 0.915677 0.916400

The following table is for what I call Pay Table 90. The feature probability is 10.2%. The bottom row shows the return for all number of picks. The greatest return is 90.94% for the pick 9.

### Pay Table 90

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 2 2 0 0 0 0 0 0
3 52 8 3 3 1 0 0 0
4 114 12 7 3 2 1 1
5 674 83 39 19 10 6
6 1400 436 102 56 33
7 2500 1800 430 159
8 2500 3400 1000
9 6000 3500
10 6000
Return 0.897283 0.903244 0.903964 0.905565 0.906626 0.906650 0.909446 0.907132

The following table is for what I call Pay Table 88. The feature probability is 10.0%. The bottom row shows the return for all number of picks. The greatest return is 88.63% for the pick 9.

### Pay Table 88

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 2 2 0 0 0 0 0 0
3 51 8 3 3 1 0 0 0
4 110 12 7 3 2 1 1
5 664 78 37 18 10 6
6 1400 435 101 55 32
7 2500 1800 420 153
8 2500 3250 1000
9 6000 3500
10 6000
Return 0.875609 0.881742 0.885121 0.883775 0.885375 0.884893 0.886328 0.886053

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