There are 10 different 3-card royal flush draws: AKQ, AKJ, AKT, AQJ, AQT, AJT, KQJ, KQT, KJT, and QJT. For today’s discussion, we are looking at these combinations in games without wild cards where you get your money back for a pair of jacks, queens, kings, or aces.
These 3-card royals are not all equal in value. There are some combinations that are always superior to other combinations regardless of the pay schedule. There are others where “it depends.”
The two biggest determinants of the value of these combinations are the rank of the highest card and the number of high cards. High cards (sometimes called “pay cards”) are those where you get your money back if you get a pair of them. In the examples here, ace, king, queen, and jack are all high cards. A ten is not a high card. To begin by using these criteria only, we have the following five categories:
1. Queen high, 2 high cards — QJT
2. King high, 3 high cards — KQJ
3. King high, 2 high cards — KQT, KJT
4. Ace high, 3 high cards — AKQ, AKJ, AQJ
5. Ace high, 2 high cards — AKT, AQT, AJT
Within each of these categories all members have the same value. That is, whatever value AKT has, AQT and AJT have exactly the same value.
When comparing two categories that each have the same number of high cards, the one with the lower highest card rank is always more valuable. Hence category 1 is more valuable than category 3 because the queen is lower than the king. Similarly category 3 is more valuable than category 5 and category 2 is more valuable than category 4. The reason for this pattern is that the lower the highest card’s rank, the more straights and straight flushes that are possible.
When two categories have the same highest card, the one with the most high cards is more valuable. Hence, category 2 is always more valuable than category 3 and category 4 is always more valuable than category 5. Another way to say this is: If a combination has a ten in it, it is less valuable than otherwise similar combinations that do not include a ten.
We cannot say a priori however, whether the combination in category 1 is higher in value than the combination in category 2; nor can we say whether the two combinations in category 3 are higher in value than the three combinations category 4. These decisions are determined by the pay schedules of the particular games being played.
Here’s a quiz: Given the following hands, which ones are affected by the particular pay schedule in determining whether QJT is more or less valuable than KQJ (Or, similarly, whether AKQ is more or less valuable than KQT?): Royal flush? Straight Flush? 4-of-a-kind? Full House? Flush? Straight? 3-of-a-kind? Two Pair? Jacks or Better? There are exactly four hands on this list that matter in making this determination. You’ll get more out of the exercise if you try to figure out which ones they are yourself before you read on to my answer.
Perhaps surprisingly, the value of the royal flush does not matter here. Each of these ten combinations requires two perfect cards to become a royal flush. There are 1,081 unique ways to draw two perfect cards. Whether you have a 1-in-1,081 chance to end up with 4,000 coins or, say, 6,000 coins affects the absolute value of each of the combinations equally. So if the royal flush increased enough to make the value of QJT greater by, say, exactly $5, then the value of KQJ would also be greater by exactly $5. Since they both increased by the identical value, their relative values with respect to each other remains unchanged.
The value of the straight flush does matter because the different categories have different straight flush potentials. Category 1 may be part of two different straight flushes (KQJT9 and QJT98), category 2 and 3 combinations may only be part of KQJT9, and the category 4 and 5 combinations may not be part of any straight flush. Therefore, a value of 400 for the straight flush instead of the more usual 250 increases the value of QJT more than it affects the value of KQJ.
The values for four-of-a-kind and full house are irrelevant to this discussion. If you hold three unpaired cards (as in our 3-card royal flush draws), you can’t end up with either of these hands.
Changes in the value of flushes affect these combinations in the opposite direction from changes in the straight flush. That is, an increase in the flush from, say, 25 to 30, increases the value of KQJ more than it increases the value of QJT. The reason for this is a bit subtle. All of the 3-card combinations are suited with each other. When you draw two cards to any kind of 3-card flush, you have 48 ways (out of 1,081 possibilities) to complete the total of royal flushes, straight flushes, and regular flushes. There is only one possible royal flush for each of our 3-card royal flush draws. From QJT you have two possible straight flushes so that means you have 45 possible flushes (48 – 1 RF – 2 SF = 45 FL). From KQJ you only have one possible straight flush so that means 46 possible flushes. Since KQJ provides more possible chances for a flush than QJT, a change in the value of the flush affects the value of KQJ more than it affects QJT.
Changes in the value of straights affect these combinations in the same direction as changes in the value of straight flushes because whenever you can get more straight flushes, you can get more regular straights as well. For example, from Category 1, there are 45 possible straights. From categories 2 and 3 there are 30 possible straights. From categories 4 and 5 there are only 15 possible straights.
The value of three-of-a-kind and two pair are irrelevant to the discussion simply because from all of these combinations you have the same chances to get these hands. So whether two pair pays 5 or 10 affects all of these combinations equally.
The value you receive for a single high pair theoretically matters because categories 1, 3, and 5 all include a ten while the other categories don’t, and pairing a ten doesn’t get you your money back while pairing up any of the other cards in the combination does. From a practical standpoint, however, on the games we’re talking about you always receive 5 coins for one of these high pairs. Basically, changes to the pay schedule for this hand don’t happen.
The question now becomes whether what I’m talking about is merely an intellectual exercise or whether there is a practical application. I suggest this really matters, particularly in understanding strategy changes for different pay tables in the same games, as well as strategy changes between games.
For those of us who play more than one video poker game, we need to use a different strategy for each game. There are a number of games (some pay schedules for Double Bonus and Double Double Bonus, for example) where KQJ and QJT are more valuable than a high pair while the other 3-card royal flushes are lower in value.
In other games (usually those where flushes return 6 for 1 while two pair returns 1 for 1), the combinations in category 5 are less valuable than a 4-card flush while the combinations in the other categories are more valuable than a 4-card flush.
There are examples from other games as well, but these are enough for now. Memorizing a strategy is a lot easier if you know why certain relationships hold. At least for me, the more I understand how similar combinations are different from each other, the easier it is to memorize a strategy. I don’t do so well with just plain rote memorization and so I avoid it if I can. I do much better if I can put things in context and understand how similar card combinations change in value based on the particular game or pay table I am playing.