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# Ace on the Deal

## Introduction

Ace on the Deal is a video poker game by Bally. I have seen it at both the Suncoast and Red Rock in November 2006. The rules are the same as normal video poker, except the player has the option to pay a sixth coin for the first card on the deal to be guaranteed to be an ace. The sixth coin also pumps up the win on the royal flush, straight flush, and four aces wins only.

There are two plausible ways the game could be programmed. The first is to deal the first card from the four aces and the other four from the remaining 51 cards. The second is to deal random hands, unseen by the player, until one is found containing at least one ace. The rules on the game do not state which method is followed, although I strongly feel the player has the right to know this rule and it should be disclosed. To test which method is used I went over to the Red Rock and to play the game, keeping track of the number of aces observed on each deal. Here are my results.

### Red Rock Experiment

Aces Observations
4 0
3 1
2 11
1 91
Total 103

The next table shows the number of combinations according to both methods of dealing.

### Ace on the Deal Number of Combinations on the Deal

Aces First Card Ace Deal until Ace
4 48 48
3 3384 4512
2 51888 103776
1 194580 778320
Total 249900 886656

The next table shows the probabilities for each number of aces on the deal under both methods.

### Ace on the Deal Probabilities on the Deal

Aces First Card Ace Deal until Ace
4 0.000192 0.000054
3 0.013541 0.005089
2 0.207635 0.117042
1 0.778631 0.877815
Total 1 1

The next table compares the actual observations with expectations based on 103 hands played and both methods of dealing.

### Ace on the Deal Observations vs. Expectations

Aces Observations First Card Ace Deal until Ace
4 0 0.019784 0.005576
3 1 1.394766 0.524145
2 11 21.386411 12.055327
1 91 80.19904 90.414952
Total 103 103 103

It is easy to eyeball that the actual observations much more closely match the "Deal until Ace" method of dealing. Doing a chi-squared test against both methods the probability of results as skewed or more against the "First Card Ace" method is 8.47%, and against the "Deal until Ace" method 91.14%.

Now that I have hopefully made a case for how the cards are dealt here is the pay table for the "9/5" Double Double Bonus game at the Suncoast, as seen on November 19, 2006.

### "9/5" Double Double Bonus Pay Table

Hand 1 Coin 2 Coins 3 Coins 4 Coins 5 Coins 6 Coins
Royal Flush 250 500 750 1000 4000 4799
Straight Flush 50 100 150 200 250 250
Four A + 2-4 400 800 1200 1600 2000 3200
Four 2-4 + A-4 160 320 480 640 800 800
Four A 160 320 480 640 800 2000
Four 2-4 80 160 240 320 400 400
Four 5-K 50 100 150 200 250 250
Full House 9 18 27 36 45 45
Flush 5 10 15 20 25 25
Straight 4 8 12 16 20 20
Three of a kind 3 6 9 12 15 15
Two pair 1 2 3 4 5 5
Pair 1 2 3 4 5 5
Nonpaying hand 0 0 0 0 0 0

The next table shows the return for the above "9/5" pay table. Unlike most of my video poker return tables the pays column is for a max coin bet. The return column is in units, based on a max coin bet. The lower right cell shows a return of 98.01%.

### "9/5" Double Double Bonus Return Table

Hand 6 Coins Pays Combinations Probability Return
Royal Flush 4799 369773856 0.000054 0.043491
Straight Flush 250 286037892 0.000042 0.001753
Four A + 2-4 3200 1236649212 0.000182 0.096987
Four 2-4 + A-4 800 588918024 0.000087 0.011547
Four A 2000 3475480848 0.000511 0.170357
Four 2-4 400 1521383688 0.000224 0.014915
Four 5-K 250 6361019412 0.000935 0.038975
Full House 45 46570657392 0.006848 0.051362
Flush 25 77043122532 0.011329 0.047205
Straight 20 45953514684 0.006757 0.022525
Three of a kind 15 467039399340 0.068678 0.171696
Two pair 5 658549186308 0.09684 0.0807
Pair 5 1865645999352 0.274344 0.22862
Nonpaying hand 0 3625739947380 0.533167 0
Total 6800381089920 1 0.980132

At the Red Rock a "7/5" Double Double pay table was used. For the six coins bet wins for a royal flush, straight flush, four aces + 2-4, and four aces + 5-K were progressive. The following table shows the return for the non-progressive wins.

### "7/5" Double Double Bonus Progressive Return Table

Hand 6 Coins Pays Combinations Probability Return
Royal Flush ? 347515680 0.000051 ?
Straight Flush ? 397951068 0.000059 ?
Four A + 2-4 ? 1224675948 0.00018 ?
Four 2-4 + A-4 800 592059432 0.000087 0.011608
Four A ? 3435376812 0.000505 ?
Four 2-4 400 1539773928 0.000226 0.015095
Four 5-K 250 6415969800 0.000943 0.039311
Full House 35 45975897216 0.006761 0.039438
Flush 25 79427550792 0.01168 0.048666
Straight 20 52474248996 0.007716 0.025721
Three of a kind 15 466607907240 0.068615 0.171537
Two pair 5 655711428312 0.096423 0.080352
Pair 5 1859530174884 0.273445 0.227871
Nonpaying hand 0 3626700559812 0.533308 0
Total 6800381089920 1 0.659600

The lower right cell shows the non-progressive wins contribute 65.96% to the return. For a 25-cent game add to this return 0.0000511024*(royal flush win in dollars) + 0.0000585189*(straight flush win in dollars) + 0.0001800893*(four aces + 2-4 in in dollars) + 0.0005051742*(four aces + 5-K win in dollars). At the time I was there on November 19, 2006, at about noon, the progressive wins were \$1199.75 for a royal, \$62.50 for a straight flush, \$819.69 for four aces + 2-4, and \$502.82 for four aces + 5-K. At this moment the return was 97.07%, following non-progressive "7/5" strategy.

Also see Deuce on the Deal.

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