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# Super Times Pay Keno

## Introduction

Super Times Pay Keno plays like ordinary four-card spot keno, except adds random multipliers. For the multiplier feature, the player must add 20% to his bet.

## Rules

1. The player picks 2 to 8 numbers in the range of 1 to 80 on four different cards.
2. The game shall randomly determine if the player wins a multiplier. In Super Times Pay video poker this probability is 1 in 15.
3. If the player does win a multiplier, it can be 2x, 3x, 4x, 5x, 8x, or 10x.
4. The game shall draw 20 numbers without replacement from balls numbered 1 to 80.
5. The base win shall be according to the number of balls that match the numbers chosen by the player. An example pay table is below.
6. If the player did not win a multiplier, then his actual win is the base win. If the player did win a multiplier, then his actual win is the product of the multiplier and the base win.

The following pay table was taken from VideoPoker.com. They invariably choose the most liberal setting for their games. Actual pay tables will vary and will probably pay less.

### Pay Table

Catch Pick 2 Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8
0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
2 15 2 2 0 0 0 0
3   45 5 3 3 1 0
4     85 13 4 2 2
5       765 77 28 12
6         1250 437 107
7           2500 1600
8             2500

## Example

In the image above I picked six numbers on each card. My bet was 10 credits per card, 8 of which went to the base bet and 2 to pay for the multiplier feature.

The game awarded a random multiplier of 3x.

On cards A and B, I caught 3 numbers for a base win of 3. Given by base bet of 8 credits and the multiplier of 3, the total win for that card was 8 × 3 × 3 = 72 for each card.

On card C, I caught 4 numbers for a base win of 4. Given by base bet of 8 credits and the multiplier of 4, the total win for that card was 8 × 4 × 3 = 96.

The total "win" for all four cards was 72+72+96 = 240. Keep in mind I bet 40 credits, so the actual win was 200.

## Analysis

The following table shows the number of combinations for catching any possible number of balls (left column) by number of picks (top row).

### Combinations Table

Catch Pick 2 Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8
0 1,770 34,220 487,635 5,461,512 50,063,860 386,206,920 2,558,620,845
1 1,200 35,400 684,400 9,752,700 109,230,240 1,001,277,200 7,724,138,400
2 190 11,400 336,300 6,501,800 92,650,650 1,037,687,280 9,512,133,400
3   1,140 68,400 2,017,800 39,010,800 555,903,900 6,226,123,680
4     4,845 290,700 8,575,650 165,795,900 2,362,591,575
5       15,504 930,240 27,442,080 530,546,880
6         38,760 2,325,600 68,605,200
7           77,520 4,651,200
8             125,970
Total 3,160 82,160 1,581,580 24,040,016 300,500,200 3,176,716,400 28,987,537,150

The following table shows the probability of catching any possible number of balls (left column) by number of picks (top row).

### Probability Table

Catch Pick 2 Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8
0 0.560127 0.416504 0.308321 0.227184 0.166602 0.121574 0.088266
1 0.379747 0.430867 0.432732 0.405686 0.363495 0.315193 0.266464
2 0.060127 0.138754 0.212635 0.270457 0.308321 0.326654 0.328146
3   0.013875 0.043248 0.083935 0.129820 0.174993 0.214786
4     0.003063 0.012092 0.028538 0.052191 0.081504
5       0.000645 0.003096 0.008639 0.018303
6         0.000129 0.000732 0.002367
7           0.000024 0.000160
8             0.000004
Total 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

The following table shows the contribution to the return for catching any possible number of balls (left column) by number of picks (top row). The bottom row shows the overall return, before considering the Super Times Pay feature.

### Base Return Table

Catch Pick 2 Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8
0 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
2 0.901899 0.277507 0.425271 0.000000 0.000000 0.000000 0.000000
3   0.624391 0.216239 0.251805 0.389459 0.174993 0.000000
4     0.260388 0.157200 0.114152 0.104382 0.163007
5       0.493367 0.238364 0.241878 0.219631
6         0.161231 0.319918 0.253238
7           0.061006 0.256728
8             0.010864
Total 0.901899 0.901899 0.901899 0.902373 0.903206 0.902177 0.903469

The rule screens at VideoPoker.com do not indicate the probability of catching a multiplier nor the average multiplier when you do. In Super Times Pay video poker, the probability of winning a multiplier is 1 in 15. In that game, the player must increase his bet by 20% to invoke the Super Times Pay feature. In that game, the average multiplier is 4.05. Doing some simple algebra, in the video poker version of the game, the feature increases the return by 0.27778%.

In Super Times Pay keno, the player must increase his bet by 25% to invoke the feature. To keep numbers round, I am going to assume the same multiplier probability of 1/15 but increase the average multiplier to 4.8. After some algebra, this results in an increase in the return by 0.26667%. The average overall multiplier, including those 14/15 times when it is zero is (14/15)×1 + (1/15)×4.8 = 1.253333.

All that said, the following table shows the overall return of the game. The return for any given cell is the product of the probability of winning, base win, 1.253333, and 4/5.

### Overall Return Table

Catch Pick 2 Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8
0 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
2 0.904304 0.278247 0.426405 0.000000 0.000000 0.000000 0.000000
3   0.626056 0.216816 0.252477 0.390497 0.175460 0.000000
4     0.261083 0.157620 0.114456 0.104660 0.163442
5       0.494683 0.239000 0.242523 0.220217
6         0.161661 0.320771 0.253914
7           0.061169 0.257413
8             0.010893
Total 0.904304 0.904304 0.904304 0.904779 0.905614 0.904583 0.905878

The bottom row shows the overall return ranges from 90.43% to 90.59%.