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Bob Dancer writes a video poker column for beginners to experts. He also writes a column with Jeffrey Compton, "Player's Edge", featuring information on promotions at various Las Vegas casinos. Player's Edge is published each Friday in the Neon section of the Las Vegas Review-Journal. Click here to send Bob Dancer an e-mail.

June 24, 2003

Drawing to a suited KQ

If you draw three cards to a suited KQ, which is more likely: a royal flush or a K-high straight flush? In games with no wild cards, like Jacks or Better or Double Double Bonus Poker, both jackpots are equally likely. In both cases you need three perfect cards to complete the hands. When you draw three cards there are 16, 215 different combinations of cards you may draw, so getting either the royal or the straight flush is a 1-in-16,215 proposition.

In Deuces Wild, however, things are more complicated. You still need three perfect cards to complete the natural royal, but there are four wild cards in the deck that affect the situation. A wild royal flush is now possible, but this will receive considerably less than a natural royal flush. A "wild" straight flush receives the same return as a "natural" straight flush.

Another complication is that frequently a hand results in BOTH a wild royal flush and a K-high straight flush, such as when you draw three deuces to a suited KQ. In these cases, you are just paid for the wild royal.

Another complication is that we are used to considering all deuces as being alike. But when we are counting combinations out of 16,215 the suits of the deuces matter. Ending up with a suited KQ9 and two black deuces, for example, counts as a different combination than a suited KQ9 and two red deuces.

First let's count the K-high straight flushes. Upon consideration, you'll notice that you MUST draw the suited nine, but the other two cards may either be natural or deuces.

Case 1: No deuces: Here we draw the JT9 of the correct suit. There is only one way to do this.

Case 2: 1 deuce. --- Here we draw the suited J9 with a deuce or the suited T9 with a deuce. Since there are four different deuces we could draw paired with each of the two natural combinations, there are a total of 8 of these.

Case 3: 2 deuces --- Here we draw the suited 9 along with two deuces. There are actually six unique ways to draw two deuces: clubs and diamonds; clubs and hearts; clubs and spades; diamonds and hearts; diamonds and spades; and hearts and spades.

Since the three cases are mutually exclusive, we add 1 + 8 + 6 = 15 chances (out of 16,215) to end up with the straight flush.

Now lets look at the chances for drawing a wild royal. Here we must have AT LEAST one deuce, and the natural cards must be some combination of A, J, and T of the correct suit. Let's take these in order.

Case 1: 1 deuce --- Here we draw AJ, AT, or JT along with any one of the four deuces for a total of 12 combinations.

Case 2: 2 deuces --- Here we draw A, J, or T along with any two deuces. Since we showed above that there are six distinct ways to end up two deuces, there are a total of 18 of these combinations.

Case 3: 3 deuces --- There are four ways to draw three deuces. One way to count them is: clubs diamonds hearts; clubs diamonds spades; clubs hearts spades; and diamonds hearts spades. An equivalent way to count them is (all except spades), (all except hearts), (all except diamonds), and (all except clubs).

Adding these three cases together we get 12 + 18 + 4 = 34 (out of 16,215) combinations.

One way to check our math is to go to any version of Deuces Wild in WinPoker, go to "Analyze", and then "Any Hand". You then see a 4 x 13 grid of possible cards to select, so we choose perhaps Kh Qh 9s 7d 4c. Whether this is a holdable hand or not depends on the pay schedule, but we can go across and see that we will result in 15 straight flushes, 34 wild royal flushes, and a single natural flush --- along with numerous other hands such as straights, flushes, full houses, 4-of-a-kinds, and, of course, the very frequent "nada" which means no scoring hand at all.

A few weeks ago I gave this problem to two different married couples at the same time. One of the couples dug in and starting working on the problem. With a couple of false starts, they eventually came up with the correct answer. The other couple said, "What good does knowing how to count hands possibly do me?" I think that's an interesting question, and I'll address it next week.