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.
One 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 ‘K’. 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 eight 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 let us 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 can 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 the correct A, J, or T along with any two deuces. Since we showed above that there are six distinct ways to end up with 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 Video Poker for Winners, go to “Analyze”, and then “Select Specific Cards”. You then see a 4 x 13 grid of possible cards to select, so we choose perhaps K♥ Q♥ 9♠ 7♦ 4♣. Whether this is a combination that is eligible to be held 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 “No Win” which means no scoring hand at all.
I understand things like this aren’t important to everybody, but they are to me. I personally like to know where the numbers in the software come from. I never know when a promotion is going to come up that pays some sort of bonus for specific hands. When it does, I want to have some depth of understanding.
If today’s article was too math-oriented for you, come back next week where I’ll describe what happened at the 2016 Blackjack Ball.