# Video Poker - General Questions

Update: The Custom Strategy Cards business no longer exists.

### Jacks or Better - Worst Possible Player

Hand | Payoff | Number | Probability | Return |
---|---|---|---|---|

Royal flush | 800 | 48564 | 0.000000 | 0.000002 |

Straight flush | 50 | 2058000 | 0.000000 | 0.000005 |

4 of a kind | 25 | 38040380 | 0.000002 | 0.000048 |

Full house | 9 | 292922028 | 0.000015 | 0.000132 |

Flush | 6 | 336550092 | 0.000017 | 0.000101 |

Straight | 4 | 6239759724 | 0.000313 | 0.001252 |

3 of a kind | 3 | 12510891616 | 0.000628 | 0.001883 |

Two pair | 2 | 34968642984 | 0.001754 | 0.003509 |

Jacks or better | 1 | 334574728656 | 0.016785 | 0.016785 |

Nothing | 0 | 19544266875156 | 0.980487 | 0.000000 |

Total | 19933230517200 | 1.000000 | 0.023717 |

There is my video poker game to play, which will correct you when you make mistakes.

There is my hand analyzer to analyze how to play any hand.

There is my game return calculator to determine the return of any game and pay table.

Finally, there is my strategy maker to create a strategy for any game.

Note: This answer was updated Nov. 13, 2013.

Loosely stated, skewness is going to correlate with how often you get a win in a session. In Jacks or Better, for the most part, you are not going to get a winning session over a few hours if you do not hit a royal. You can sit down at Double Double Bonus and be a winner after a few hours more often because of the big quad payouts. Because most people are subject to cognitive biases, the pain from a loss is twice the pleasure from a win. People do not really play Double Double Bonus because they like the variance, they play because they have a greater shot at winning. The following table shows some key statistics for four common video poker games. It is interesting to note that skew is greatest for Jacks or Better.

### Key Video Poker Statistics

Statistic | JoB — 9/6 | BP — 8/5 | DDB — 9/6 | DW — NSUD |
---|---|---|---|---|

Return | 0.995439 | 0.99166 | 0.989808 | 0.997283 |

Variance | 19.514676 | 20.904113 | 41.985037 | 25.780267 |

Skew | 147.114643 | 134.412152 | 66.495372 | 101.23991 |

(Excess) Kurtosis | 26,498 | 23,202 | 6,679 | 14,550 |

JoB — 9/6 = Full pay Jacks or Better

BP — 8/5 = Standard pay Bonus Poker

DDB — 9/6 = Standard pay Double Double Bonus Poker

DW — NSUD = "Not so Ugly Ducks" Deuces Wild

How can knowing this actually help the video poker player? I suppose one could say that a game with a large skew has a greater chance of a loss over a session of a few hours. For example, in Jacks or Better, if you don’t hit any royals, the house edge will probably eventually grind your bankroll down. However in a game like Deuces Wild or Double Double Bonus, the second highest wins can pull you out of the hole over a session. In other words, the skew keeps you from winning when you are not hitting royals. Knowing the skew won’t increase your odds, but it is mentally helpful to know what to expect. So, the next time you take a beating in 9/6 Jacks, blame it on the skew.

My thanks to Jeff B. for his help with this question.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

Let's start by looking at some no-nonsense video poker games. The following table shows the probability of a royal, from highest to lowest. This table does not count wild royals, which pay much less than a natural.

### Video Poker Royal Probability

Game | Probability | Inverse |
---|---|---|

17-7 Joker Poker | 0.0000259892 | 38,478 |

8-5 Bonus Poker | 0.0000248551 | 40,233 |

9-6 Jacks | 0.0000247583 | 40,391 |

9-6 Double Double Bonus | 0.0000245102 | 40,799 |

9-6 Bonus Poker Deluxe | 0.0000237661 | 42,077 |

"Full pay" deuces wild | 0.0000220839 | 45,282 |

10-7 Double Bonus | 0.0000208125 | 48,048 |

Surprisingly, Joker Poker is the most likely to get a natural royal. This surprises me because of the extra card, which can't be used to make a natural royal.

Next, let's look at some non-standard video poker games.

In 9-6 Jacks Royal Draw, the odds are 1 in 12,178.

In 7-5 Jacks Second Chance Royal, the odds are 1 in 10,827. This includes "second chance" royals, which pay 200 only.

Finally, I think an honorable mention should be made for Triple Double Bonus, where four aces plus a 2-4 kicker pay the same 800 per coin bet as a royal flush. The odds of either 800 win are 1 in 10,823, based on the 9-7 pay table. Even better is Royal Aces Bonus Poker, which pays 800 for any royal flush or four aces, for a probability of 1 in 3,673 for a win of 800.

However, all things considered, my answer for the game with the most natural royals is Chase the Royal. Based on the 9-6 Jacks pay table, the probability of a royal flush is 1 in 9,084.

### "9-6" Jacks or Better

Hand | Payoff | Combinations | Probability | Return |
---|---|---|---|---|

Royal flush | 800 | 493512264 | 0.00002476 | 0.01980661 |

Straight flush | 50 | 2178883296 | 0.00010931 | 0.00546545 |

Four of a kind | 25 | 47093167764 | 0.00236255 | 0.05906364 |

Full house | 9 | 229475482596 | 0.01151221 | 0.10360987 |

Flush | 6 | 219554786160 | 0.01101451 | 0.06608707 |

Straight | 4 | 223837565784 | 0.01122937 | 0.04491747 |

Three of a kind | 3 | 1484003070324 | 0.07444870 | 0.22334610 |

Two pair | 2 | 2576946164148 | 0.12927890 | 0.25855780 |

Jacks or Better | 1 | 4277372890968 | 0.21458503 | 0.21458503 |

Nothing | 0 | 10872274993896 | 0.54543467 | 0 |

Total | 19933230517200 | 1 | 0.99543904 |

Most of my video poker return tables for 52-card games have the same number of combinations of 19933230517200. The question is why?

First, there are combin(52,5) = 2,598,960 ways to choose five cards out of 52.

Second, there are up to combin(47,5) = 1,533,939 combinations on the draw, depending on how many cards the player discards. The following table shows the number of draw combinations in the second column according to the number of discards.

### Draw Combinations

Discard | Combinations | Weight | Product |
---|---|---|---|

0 | 1 | 7,669,695 | 7,669,695 |

1 | 47 | 163,185 | 7,669,695 |

2 | 1,081 | 7,095 | 7,669,695 |

3 | 16,215 | 473 | 7,669,695 |

4 | 178,365 | 43 | 7,669,695 |

5 | 1,533,939 | 5 | 7,669,695 |

The least common multiple of the numbers in the second column is 7,669,695. This number can be expressed as 5×combin(47,5). To keep the total number of combinations the same for every hand on the deal I weight the draw combinations so that the total combinations on the draw is 7,669,695.

So, 19,933,230,517,200 = combin(52,5)×combin(47,5)×5. Some of my 52-card video poker tables have a smaller number of combinations. This is because sometimes the total number of combinations for each hand in the final return table has a greatest common divisor greater than one. In this case I sometimes divide each total by whatever the greatest common divisor is. My video poker analyzer does this automatically.

This question was raised and discussed at my Wizard of Vegas forum.

Yes, that machine is no longer. As you point out, the previous number two game now bubbles up to the number one spot. That is a 100.92% Double Deuces game at the El Cortez. The full pay table is 940-400-25-16-13-4-3-2-2-1. It can be found on the slot room on the north end of the casino full of old cathode raw machines. Here is a picture of me standing next to it.

In case you're wondering, the difference between Loose Deuces and Double Deuces is the former pays 500 for four deuces and the latter 400.

For example, in the 97.19% pay table for two pair joker poker and a Qh, 10d, 5d, 7c, and 2c on the draw, the best play is a tie between holding only the 10d, 5d, and 7c, with an expected value of 0.240703 for all three. This can be verified using my video poker hand analyzer.

Here are other such plays for the same joker poker pay table:

- QC 10S 5D 7H 2C
- QH 10D 5D 7H 2C
- KH 10D 5D 8C 3C
- KC 10S 5D 8H 3C
- KH 10D 5D 8H 3C

My thanks to Gary Koehler for his help with this question.

But the methodology page only deals with a standard 52-card deck with no wild cards. I have tried to expand the algorithm on my own to allow adding one to four jokers to the deck, but so far the results have been slow and inaccurate. Part of the problem is that wild cards seem to add a lot of complexity as my algorithm substitutes 52 possibilities for the wild cards and furthermore I don't think having duplicate indexes is allowed by the algorithm. I was wondering if the methodology page could be expanded to give tips for dealing with wild card situations in a similarly efficient way that it already deals with the standard deck.

I love the website and want to thank you for the assistance it has provided to this point, as I don't think I could have completed my project without it. Have a great day!

Cycling through all 52 ways to use a wild card will definitely be slow, especially if there are multiple wild cards. Here is how I score a hand with wild cards:

- If you have a natural royal flush, then score the hand as such.
- Otherwise, count the number of wild cards in the hand.
- Then score the hand according to this number and the value of the other cards.

For example, suppose you know you have two wild cards. You could use this pseudo-code to score the hand based on the other three cards:

- If the three cards are suited and the lowest one is at least a ten, then you have a wild royal.
- Otherwise, if the three cards are suited and the difference between the highest and smallest is four or less, then you have a straight flush.
- Otherwise, if the three cards are suited, the highest is an ace, and the second highest is a five or less, then you also have a straight flush (A2345).
- Otherwise, if you have a pair, then you have a four of a kind.
- Otherwise, if the three cards are suited, you have a flush.
- Otherwise, if the difference between the highest and smallest is four or less, then you have a straight.
- Otherwise, if the highest is an ace, and the second highest is a five or less, then you also have a straight (A2345).
- Otherwise, you have a pair.

You already have code for no wilds, so you'll have to write separate code for one to either four or five wilds, depending on whether you may wish to analyze Deuces and Joker Wild.

The first step is to modify my calculator to include a line item for all 13 four of a kinds. Here is that modified return table:

### Modified Jacks or Better Return Table

Event | Pays | Combinations | Probability | Return |
---|---|---|---|---|

Royal flush | 800 | 493,512,264 | 0.000025 | 0.019807 |

Straight flush | 50 | 2,178,883,296 | 0.000109 | 0.005465 |

Four A | 25 | 3,900,253,596 | 0.000196 | 0.004892 |

Four K | 25 | 3,904,533,816 | 0.000196 | 0.004897 |

Four Q | 25 | 3,898,370,196 | 0.000196 | 0.004889 |

Four J | 25 | 3,886,872,684 | 0.000195 | 0.004875 |

Four 10 | 25 | 3,471,687,732 | 0.000174 | 0.004354 |

Four 9 | 25 | 3,503,226,684 | 0.000176 | 0.004394 |

Four 8 | 25 | 3,504,128,652 | 0.000176 | 0.004395 |

Four 7 | 25 | 3,504,825,252 | 0.000176 | 0.004396 |

Four 6 | 25 | 3,504,861,888 | 0.000176 | 0.004396 |

Four 5 | 25 | 3,504,895,944 | 0.000176 | 0.004396 |

Four 4 | 25 | 3,504,032,676 | 0.000176 | 0.004395 |

Four 3 | 25 | 3,503,177,148 | 0.000176 | 0.004394 |

Four 2 | 25 | 3,502,301,496 | 0.000176 | 0.004393 |

Full house | 9 | 229,475,482,596 | 0.011512 | 0.103610 |

Flush | 6 | 219,554,786,160 | 0.011015 | 0.066087 |

Straight | 4 | 223,837,565,784 | 0.011229 | 0.044917 |

Three of a kind | 3 | 1,484,003,070,324 | 0.074449 | 0.223346 |

Two pair | 2 | 2,576,946,164,148 | 0.129279 | 0.258558 |

Jacks or better | 1 | 4,277,372,890,968 | 0.214585 | 0.214585 |

Nothing | 0 | 10,872,274,993,896 | 0.545435 | 0.000000 |

Total | 19,933,230,517,200 | 1.000000 | 0.995439 |

The probability of getting any four of a kind is 0.002363.

The next question to be answered is how many four of a kinds will take on average to get all 13 kinds? To answer that question, I created my Expected Trials Calculator. To use it, enter the number of combinations of each four of a kind in the first 13 cells. The calculator will tell you that it will take an expected 41.532646 four of a kinds to get all 13 kinds.

So, the expected number of hands played to get all 13 four of a kinds is 41.341739/0.002363 = 17,580.

Let's look at 8-5 Bonus Poker as an example. The following table shows the return by number of lines bet.

- 4 lines: 99.375%
- 3 lines: 99.279%
- 2 lines: 99.214%
- 1 line: 99.166%

The next list shows the cost to not playing the maximum lines according to how many lines are played.

- 4 lines: 0.000%
- 3 lines: 0.095%
- 2 lines: 0.160%
- 1 line: 0.209%

If the player gets the Dream Card, then the probability the other four cards will fall into this situation is 1.49%. Given the 46.7% probability of the Dream Card, this situation will happen with probability 0.70% or once every 144 hands.

Using my video poker hand analyzer, the expected value of a pair of fours in 8/5 Bonus Poker is 0.855134. The expected value of a pair of 5's to 10's is 0.813506. So the cost of the bug every time it occurs, assuming the player accepts the Dream Card, is 0.041628 in expected value.

The overall cost to the return of the game is 0.006955 × 0.041628 = 0.000290, or about 0.03%.

The following table shows the probability of having 0 to 5 cards to a royal on the deal, assuming the player always choose the suit that already has the most cards to the royal, the probability of completing the royal, and the product.

### Royal or Nothing Player

Card to Royal on Deal |
Deal Probability |
Probability Complete Royal |
Product |
---|---|---|---|

0 | 0.61538462 | 0.00000261 | 0.00000160 |

1 | 0.35444947 | 0.00003064 | 0.00001086 |

2 | 0.02835596 | 0.00070472 | 0.00001998 |

3 | 0.00173608 | 0.01057082 | 0.00001835 |

4 | 0.00007234 | 0.11627907 | 0.00000841 |

5 | 0.00000154 | 1.00000000 | 0.00000154 |

Total | 1.00000000 | 0.00006075 |

The lower right cell shows a "royal or nothing" player has a probability of making a royal flush of 0.000006075, or 1 in 16,461.

Let's assume that the game picks five random cards for the draw, which will be waiting in a queue for the player to discard. For example, if the player discards three, he will get the next three in the queue. The answer is the same, by the way, if there is a specific card on the deal assigned to each card on the draw. That said, the following table shows the probability of holding each number to the royal on the deal, the probability of completing it on the draw, and the product. The lower right cell shows an overall probability of 0.00006075, which equals 1 in 16,461.

### Perfect Peeker in Video Poker

Card to Royal on Deal |
Deal Probability |
Probability Complete Royal |
Product |
---|---|---|---|

0 | 0.61538462 | 0.00000261 | 0.00000160 |

1 | 0.35444947 | 0.00003064 | 0.00001086 |

2 | 0.02835596 | 0.00070472 | 0.00001998 |

3 | 0.00173608 | 0.01057082 | 0.00001835 |

4 | 0.00007234 | 0.11627907 | 0.00000841 |

5 | 0.00000154 | 1.00000000 | 0.00000154 |

Total | 1.00000000 | 0.00006075 |

The answer is rather involved, because there are a number of ways the player can have a royal possibility, after the deal, in multiple suits. I assume the player always keeps the cards in the suit with the greatest chance at a royal and picks arbitrarily in the event two or more suits are tied for the most cards to a royal. That said, let me define some abbreviations:

- Royal cards = cards of rank 10 to ace.
- H = Royal cards in hearts.
- S = Royal cards in hearts.
- C = Royal cards in hearts.
- D = Royal cards in hearts.
- x = Non-royal card

The following table shows the number of combinations of every possible situation. A row will include all mathematically equivalent cases. For example, Hxxxx will include having one card to a royal only in any suit (not just hearts).

### Combinations to Royal after Deal

Hand | Cards to Royal | Combinations |
---|---|---|

HHHHH | 5 | 4 |

HHHHS | 4 | 300 |

HHHHx | 4 | 640 |

HHHSS | 3 | 1,200 |

HHHSC | 3 | 3,000 |

HHHSx | 3 | 19,200 |

HHHxx | 3 | 19,840 |

HHSSC | 2 | 6,000 |

HHSSx | 2 | 19,200 |

HHSCD | 2 | 5,000 |

HHSCx | 2 | 96,000 |

HHSxx | 2 | 297,600 |

HHxxx | 2 | 198,400 |

HSCDx | 1 | 20,000 |

HSCxx | 1 | 248,000 |

HSxxx | 1 | 744,000 |

Hxxxx | 1 | 719,200 |

xxxxx | 0 | 201,376 |

Total | 2,598,960 |

The next table shows the overall probability of having 0 to 5 cards to a royal after the deal.

### Cards to Royal Probabilities

Cards to Royal | Probability |
---|---|

5 | 0.0002% |

4 | 0.0362% |

3 | 1.6637% |

2 | 23.9403% |

1 | 66.6113% |

0 | 7.7483% |

Total | 100.0000% |

Not that you asked, but if a player followed a "royal or nothing" strategy, his probability of a royal per hand would be 1 in 23,162.

As a basis of comparison, the probability of a royal flush in conventional video poker ranges from about 1 in 40,000 to 45,000, assuming optimal strategy. Here are some exact figures for some randomly selected games:

- 9-6 Jacks or better: 1 in 40,391
- 25-15-9-4-4-3 (Illinois) Deuces Wild — 1 in 43,423
- 9-7 Triple Double Bonus: 1 in 45,358

For Royal hungry players, the probability goes up significantly in Chase the Royal. This is an early video poker variant where the player may exchange a dealt pair of face cards for three to a royal flush on the deal. To make it a good value to trade, the game bumps up the win on the straight and flush, if you switch. The exact royal probability will depend on the base game and pay table. The probability is maximized with a base game of Triple Bonus and 8-5 pay table, at a royal frequency of 1 in 9,151. This includes both royals on the draw, which pay 800 for 1, at a frequency of 1 in 9,282 and on the deal, which pay 2000 for 1, at a frequency of 1 in 649,773.

However, it gets even better if you consider games in which the player must pay a fee equal to his base wager to enable a feature. In Draw Poker with Dream Card (not to be confused with Dream Card Poker), the player often gets the card of his dreams (assuming its a mathematician doing the dreaming) as the fifth card on the deal. The Dream Card probability of maximized in Jacks or Better, at 50.5%. In 9-6 Jacks or Better, the overall Royal frequency is 1 in 8,105. Keep in mind that with the fee to enable the feature, the effective win on a Royal drops to 400 for 1.

As long as I mention Dream Card Poker (a different game than Draw Poker with Dream Card), the Royal frequency in that game is not as high. It appears to be highest in 11-8-6 Jacks or Better at 1 in 15,034.

This answer does not consider Movin' On Up Poker, which is an old and obscure video poker game where the player gets two or three draws, instead of one. I don't know the Royal frequency in that game, but in Triple Draw, in which the player must pay a fee equal to five times his base wager to enable the extra two draws, I roughly estimate it to be about 1 in 4,000.

In conclusion, if you don't count games where the player must pay an extra fee to enable some kind of gimmick, my answer is Chase the Royal.

What is the cost in player errors if I play the optimal strategy for Not so Ugly Ducks in Illinois Deuces?

As a reminder, here are the pay tables mentioned:

Not so Ugly Ducks: 1-2-3-4-4-10-16-25-200-800.

Illinois Deuces: 1-2-3-4-4-9-15-25-200-800

Next, here is the return table for Not so Ugly Ducks, following the optimal strategy for that game.

### Not so Ugly Ducks -- Correct Strategy

Event | Pays | Combinations | Probability | Return |
---|---|---|---|---|

Natural royal flush | 800 | 458,696,304 | 0.000023 | 0.018409 |

Four deuces | 200 | 3,721,737,204 | 0.000187 | 0.037342 |

Wild royal flush | 25 | 38,006,962,464 | 0.001907 | 0.047668 |

Five of a kind | 16 | 61,961,233,656 | 0.003108 | 0.049735 |

Straight flush | 10 | 102,392,435,976 | 0.005137 | 0.051368 |

Four of a kind | 4 | 1,216,681,289,508 | 0.061038 | 0.244151 |

Full house | 4 | 520,566,943,104 | 0.026116 | 0.104462 |

Flush | 3 | 413,870,908,056 | 0.020763 | 0.062289 |

Straight | 2 | 1,142,885,476,800 | 0.057336 | 0.114671 |

Three of a kind | 1 | 5,325,911,611,716 | 0.267188 | 0.267188 |

Nothing | 0 | 11,106,773,222,412 | 0.557199 | 0.000000 |

Total | 19,933,230,517,200 | 1.000000 | 0.997283 |

Next, here is the return table for Illinois Deuces, using the correct strategy for that pay table. The lower right cell shows a return of 0.989131.

### Illinois Deuces -- Correct Strategy

Event | Pays | Combinations | Probability | Return |
---|---|---|---|---|

Natural royal flush | 800 | 459,049,128 | 0.000023 | 0.018423 |

Four deuces | 200 | 3,727,422,492 | 0.000187 | 0.037399 |

Wild royal flush | 25 | 38,117,987,136 | 0.001912 | 0.047807 |

Five of a kind | 15 | 62,201,557,608 | 0.003120 | 0.046807 |

Straight flush | 9 | 98,365,859,016 | 0.004935 | 0.044413 |

Four of a kind | 4 | 1,221,942,888,444 | 0.061302 | 0.245207 |

Full house | 4 | 522,030,131,520 | 0.026189 | 0.104756 |

Flush | 3 | 407,586,633,720 | 0.020448 | 0.061343 |

Straight | 2 | 1,145,767,137,120 | 0.057480 | 0.114961 |

Three of a kind | 1 | 5,342,397,992,292 | 0.268015 | 0.268015 |

Nothing | 0 | 11,090,633,858,724 | 0.556389 | 0.000000 |

Total | 19,933,230,517,200 | 1.000000 | 0.989131 |

The next table shows the return table using the combinations and probability from Not so Ugly Ducks on the pay table for Illinois Deuces. The lower right cell shows a return of 0.989131.

### Illinois Deuces -- NSUD Strategy

Event | Pays | Combinations | Probability | Return |
---|---|---|---|---|

Natural royal flush | 800 | 458,696,304 | 0.000023 | 0.018409 |

Four deuces | 200 | 3,721,737,204 | 0.000187 | 0.037342 |

Wild royal flush | 25 | 38,006,962,464 | 0.001907 | 0.047668 |

Five of a kind | 15 | 61,961,233,656 | 0.003108 | 0.046627 |

Straight flush | 9 | 102,392,435,976 | 0.005137 | 0.046231 |

Four of a kind | 4 | 1,216,681,289,508 | 0.061038 | 0.244151 |

Full house | 4 | 520,566,943,104 | 0.026116 | 0.104462 |

Flush | 3 | 413,870,908,056 | 0.020763 | 0.062289 |

Straight | 2 | 1,142,885,476,800 | 0.057336 | 0.114671 |

Three of a kind | 1 | 5,325,911,611,716 | 0.267188 | 0.267188 |

Nothing | 0 | 11,106,773,222,412 | 0.557199 | 0.000000 |

Total | 19,933,230,517,200 | 1.000000 | 0.989038 |

The cost of errors is the optimal return for Illinois Deuces (second table) less the return for Illinois Deuces using NSUD strategy (third table) = 0.989131 - 0.989038 = 0.000093.

I read about the Reversible Royal game with the 105.22% return in your article at Wizard of Vegas. That return assumes optimal strategy, including for card order. What is the return if I assuming an average royal win? How about if I use ordinary 6-5 Bonus Poker strategy, which is the base pay table?

Assuming no strategy deviations, 1 in 60 royals will be sequential. The reversible royal jackpot pays 161,556 for 1. Any other royal pays 800 for 1. The average royal win is thus (1/60)*161,556 + (59/60)*800 + 17,396 for 1.

If we assuming all royals pay 17,396 and play optimal strategy based on that royal win, then the return drops to 103.56%.

If we play standard 6-5 Bonus Poker strategy, which is the base pay table, then the return drops further to 101.97%.

In your video poker programming tips, you explain how that although there are 2,598,960 possible starting hands in video poker, with a 52-card deck, there are only 134,459 classes of hands necessary to analyze.

My question is how many classes are there for joker poker?

For this one, I turned to my esteemed colleague, Gary Koehler, who is an expert at video poker math. Here are his answers, according to the number of jokers:

- 1 Joker: 150,891
- 2 Jokers: 169,078
- 3 Jokers: 189,189
- 4 Jokers: 211,406
- 5 Jokers: 235,925

In your video poker programming tips, you explain how that although there are 2,598,960 possible starting hands in video poker, with a 52-card deck, there are only 134,459 classes of hands necessary to analyze. My question is if someone were playing a game where the order of the cards matter, like Ace$ Bonus Poker or one with a jackpot for a sequential royal, how many different classes of hands would be needed to analyze?

For this one, I turned to my esteemed colleague, Gary Koehler, who is an expert at video poker math. His answer is 15,019,680.

What is the average number of cards held after the draw in video poker?

The following table shows the average number of cards held in 10 different games and pay tables. The average of the games listed is 2.05.

### Average Cards Held in Video Poker

Game | Pay Table | Return | Ave. Cards Held |
---|---|---|---|

Bonus Deuces | 10-4-3-3 | 97.36% | 1.845550 |

Deuces Wild | 25-15-9-5-3 | 100.76% | 1.926010 |

White Hot Aces | 9-5 | 99.57% | 2.055630 |

Super Double Double Bonud | 7-5 | 99.17% | 2.057280 |

Double Double Bonus | 9-5 | 97.87% | 2.058390 |

Triple Double Bonus | 8-5 | 95.97% | 2.072620 |

Bonus Poker | 8-5 | 99.17% | 2.080610 |

Jacks or Better | 9-5 | 98.45% | 2.081030 |

Bonus Poker Deluxe | 8-5 | 97.40% | 2.150470 |

Double Bonus | 9-6-5 | 97.81% | 2.173550 |

For 9-6 Jacks or Better you will win $1200 or more with every four of a kind or higher. The return for a full house and less is 0.911103. You'll have to churn though your original bankroll 1/(1-0.911103) = 11.249016 times until your cash has either been converted to check or lost to the house edge. The house edge is 0.004561. Thus, you can expect to lose 0.004561 × 11.249016 = 0.051306 times your original bankroll.

Interestingly, the cost to convert cash to checks is less in 9-6 Double Double Bonus, despite the higher house edge. That game has higher wins for all four of a kinds, so your checks are larger. In that game, the return from a full house and less is 0.777138. That means, you'll have to cycle through your bankroll 1/(1-0.777138) = 4.487076 times to convert it to check. The house edge in 9-6 double double bonus is 0.010192. Thus, the expected loss converting to cash is 4.487076 × 0.010192 = 0.045733.

You could cut down the cost even more with strategy deviations that target four of a kinds and higher, but I'll leave that as an exercise for the reader.

This question is asked and discussed in my forum at Wizard of Vegas.