Distinguished Royals

            I've spent the past few weeks walking through the strategy table for full-pay jacks or better video poker.  This week, we are in the heart of the table, which is to say we are in the middle of the really bad hands.  But, bad hands are the harder hands to play correctly and are just as important as the good hands in achieving the theoretical strategy of any particular game.  We finished up last week with the 4-Card Straights.  From here the hands only get uglier.  The next several entries are:

·       3-Card Double Inside Straight Flush with 2 High Cards

·       3-Card Inside Straight Flush with 1 High Card

·       3-Card Straight Flush with 0 High Cards

·       2-Card Royal Flush - "V3"

            If you're relatively new to strategy tables, this part of the table has problem just given you a doozy of a headache!  The good news is that it is not nearly as confusing as it might appear.  Many of the hands listed above cannot co-exist - meaning that you can't have more than one in a particular hand.  Thus, remembering the exact order may not be as important as it might appear to be.  For example, you can't have a 3-Card Double Inside Straight Flush with 2 High Cards in the same hand as a 3-Card Straight with 0 High Cards.  You'd either have to have at least 4 cards of one suit or have 2 sets of 3-Cards of different suits - rather difficult with 5 cards. 

            There are still important things we can learn from this section of the table.  We separate the hands the way we do because in some games the impact of the subtle differences and the order of the hands will be different and thus become pertinent.  The first thing you might notice is the relationship of the top three hands.  We get a sense of the importance of a High Card.  A 3-Card Double Inside Straight Flush with 2 High Cards has a higher expected value than a 3-Card Inside Straight Flush with only 1 High Card.  Essentially, the value of the extra High Card is greater than the value of the additional Straight Flushes (and Straights) that may occur as a result of having an Open vs. Inside vs. Double Inside Straight Flush.  What this should also tell us is that you shouldn't hold your breath for those Straight Flushes.  They will occur, but not often.  At the same time, I have written at length over the years about how the Straight Flush is the forgotten hand of video poker.  Playing 3-Card Straight Flushes correctly is very important to drawing them in proper abundance.  While their pays are far short of the Royal, they still pay double what Quads pays so their value should not be dismissed.

            Next up is the 2-Card Royal Flush - "V3".  2-Card Royals are given 4 different designation from "V0" to "V3".   We need to do this because the expected values of many of the hands in this part of the table differ by only 0.01 or 0.02.  As not all 2-Card Royals have identical expected values, we need to distinguish between them.  V3 means that the 2-Card Royal contains neither a 10 nor an Ace.  An Ace in a 2-Card Royal essentially makes it a Double Inside Royal.  All 2-Card Royals have the same number of ways to make a Royal - one.  But, with the Ace, we eliminate all ways to make a Straight Flush.  While a '10' doesn't have this problem, it does have the problem that it is not a High Card.  So, Aces are worth less than Jacks, Queens and Kings and 10's are worth less than Aces.  Thus, a 2-Card Royal with neither an Ace nor a '10' is the one with the highest expected value.  A V2 2-Card Royal means that the 2-Card Royal has an Ace, but no 10.  A V1 2-Card Royal is one that has no Ace, but does have a 10.  Lastly, a V0 2-Card Royal consists of a 10 AND an Ace.

            For the moment, I'll jump to below the strategy table - to the V0 2-Card Royal.  We are 'below' the strategy table because this hand does not exist on the strategy table for full-pay jacks or better.  This means that we DO NOT PLAY an A-10 2-Card Royal.  Barring the other three cards forming an otherwise playable hand, we would simply hold the single Ace.  More on that in a couple of weeks.

            The proper play of 2-Card Royals is critical to learning how to master video poker strategy.  Unlike the prior 3 hands, 2-Card Royals overlap with EVERYTHING.  You'll have 2-Card Royals with High Pairs, Low Pairs, 4-Card Straights, 4-Card Flushes, 3-Card Straight Flushes, 3-Card Inside Straight Flushes, etc...   If you blindly go after every 2-Card Royal, you'll hit more than your fair share of Royals, but you'll lower the payback of your play.  If you ignore 2-Card Royals, you'll miss you fair share of Royals AND lower the payback of your play.  The only answer is to play them when you are supposed to.  Based on the portion of the strategy table shown, you can easily have a 3-Card Double Inside Straight Flush with 2-High Cards AND a 2-Card Royal - V3.  For example, you could have 8-J-Q suited.  From the strategy table, we learn that we keep the 8 in this case.  You might also have a 3-card Straight Flush completely apart from a 2-card Royal.  For example, 3H, 4H, 5H, JD, QD.  In this case, we discard the 2-Card Royal in favor of the 3-Card Straight Flush with 0 High Cards.

            I've now covered about 2/3 of the strategy table for jacks or better video poker.  Next week, we continue through the rest of the messy hands. 

When 2 is better than 3

            When my father developed the first strategies for video poker, a few surprises definitely showed up.  Playing 4-Card Flushes over Low Pairs was not such a surprise, but playing the Low Pair over 4-Card Straights was.   One of the other significant surprises was how to play the numerous hands that contain High Cards.   If you had 3 High Cards of the same suit, it wasn't much of a surprise to hold all three.  Even if one of those 'High' cards was only a 10.  A 3-Card Royal is a pretty strong hand, even if it takes a bit of a long shot to actually hit the Royal.

            Without the mathematical analysis of video poker to guide the Player, most found themselves holding on to all cards Jack or Higher.  This would probably be the right play if you were sitting at a Poker table.  When playing Poker, there is little benefit to drawing a Royal over a Straight or a Flush.  All are very likely to leave you as a winner and the amount you win will not change based on your final hand value.  In the meantime, you'll increase your chance (or will you?) of grabbing a High Pair which will may be enough to win the hand.

            But video poker is not table poker and a Royal has a good deal more value than a Straight or a Flush - 200 to 130+ times as much.  This makes taking the risk of getting the Royal far more worthwhile in video poker than table Poker.  As a result, the decision of what to do when you're dealt a J♥, Q♦, A♥ not as clear as one might think.  Let's take a look at the detailed analysis.

            If the Player holds the 3 High Cards, there are 1081 possible resulting draws.  32.2% of the time the Player will wind up with a High Pair.  If the Player holds only the 2 suited High Cards, he will wind up with a High Pair 30.3% of the time.  So, the probability is a little less, but we're not talking a huge difference.  The Player may only have 2 High Cards instead of 3, but he will draw 3 cards instead of 2 helping to even things out a bit.

            Moving on, with the 3 High Cards, the Player will draw a Two Pair about 2.5% of the time.  With the 2 High Cards he will pull a Two Pair about 4.4% of the time.  The score has been quickly settled with the High Pair frequencies.  For as often as the Player will wind up with fewer Pairs he will wind up with more Two Pairs.  Given Two Pairs pay twice as much, this puts the 2 suited High Cards in the lead.            The pattern continues with Trips, with the Player drawing about twice as many by holding onto only the 2 suited High Cards.  

            Things turn around when we look at Straights.  It should be no surprise that the probability of drawing a Straight goes way up when you hold 3 High Cards as compared to 2 High Cards.   The exact probabilities will be impacted by the specific cards, but in this particular case the probability with 3 High Cards is about 1.5% vs 0.3% for 2 High Cards. 

            For the 3 High Card hands, the hands stop there.  There is ZERO chance of drawing a Flush, Full House, Quads, a Straight Flush or the Royal.  For the 2 High Card hand, we still have a 1% chance of drawing a Flush and slim, yet possible chances to get a Full House, Quads or the elusive Royal.  In this particular case, there is no chance for a Straight Flush, but if I had chosen a suited J-K for my example, this would exist as well.

            If we were to ignore all the hands Flush and above, the two hands would have nearly identical expected values, with the 3 High Card hand slightly higher, However, there is no reason to ignore these hands.  In fact, we specifically play the 2 High Card hand for the specific reason that we have the opportunity to draw all these relatively high paying hands simply by discarding the 1 off-suit card, all while barely impacting the overall expected value of the lower hands. 

            As a result, the decision is not really a hard one to make, even if it was an originally surprising part of the strategy.  Our 2-Card Royal with an Ace has an expected value of about 0.58.  Our 3 High Card expected value is a mere 0.46%. 

            This type of hand is a fairly common one and repeatedly playing it the wrong way will take a bite out of your bankroll.  This is why the 'seat of your pants' approach or using table Poker strategy can be quite ruinous to your results.  Sometimes, 2 can be better than 3.

Royal Appearance

            Last week’s column was prompted by a reader who raised some concerns that Players who use their frequent player cards are somehow cheated by casinos.  The ‘proof’ of this is that some locals (i.e. frequent Players) don’t seem to get as many Royals as the tourists.  Previously, I had cited at least two reason for this. 

            The first is selective memory.  We tend to remember things we want to remember.  When we go through an extended cold streak, every other scream of “Royal” is burned into our brains.  I’m guessing that in the week you hit your last Royal, someone else did too that week, but it didn’t bother you one bit.  If you’ve gone a year without one, everytime someone gets one, it hits you like a ton of bricks.

            Secondly, even if you’re trying to be relatively objective about it, you also have to remember that ‘you’ are greatly outnumbered by ‘them’.  Even if there are a couple of you playing together, there are dozens if not hundreds of other people playing around you.  It is no surprise that they WILL actually get more Royals than your group will.

            There is, of course, another possibility – other people ARE actually get more Royals than you are!  So, am I buying into the whole ‘rigged’ video poker machine theory?  ABSOLUTELY NOT! 

            But, the number of Royals you get over an extended period of time is greatly influenced by the strategy you use.  So, there are two possibilities.  YOU may be using the wrong strategy which is reducing the probability of a Royal OR the other guy is using the wrong strategy which might INCREASE the probability of a Royal.

            Let’s look at these two scenarios.  The proper strategy for any particular video poker machine is one that maximizes the overall payback, not one that maximizes the probability of hitting a Royal.  What do you do when you’re dealt the following?

A♥       Q♥       10♥      5♥        5♠

            Do you hold the Low-Pair?  The 4-Card Flush?  The 3-Card Royal?

            The correct answer is the 3-Card Royal.  If you’re playing one of the other two, not only are you hurting yourself from a payback perspective, you’re lowering your chances of hitting a Royal.  By the way, the decision is not even close.  The expected value (EV) of the 3-Card Royal is 1.41.  The 4-Card Flush has an EV of 1.22 and the Low Pair a meager 0.82.

            So, if you’re not playing this hand correctly, don’t be surprised if some others around you are hitting more Royals.  Of course, they may have their own issues.  What do you do when you’re dealt the following?

A♥       Q♥       10♥      5♥        Q♠

            The correct answer is hold the High Pair with an expected value of 1.54.  Now, you may be doing this, but that ‘tourist’ behind you may not.  What is the impact of holding the 3-Card Royal.  Well, he’ll lower his overall expected payback, BUT he will increase his probability of hitting a Royal.

            The examples I used here are not the most common occurrences, so these will not make a big difference to the frequency of a Royal happening.  Far more common are the hands that include a 2-Card Royal that also include 3-Card and 4-Card Flushes and Straights.  I have little doubt that there are many novice Players who get Royal fever and just play every 2-Card Royal instead of 4-Card Straights and Flushes or 3-Card Straight Flushes.  Doing so, will make them hit more Royals than you will, but they won’t be any richer for it. 

            Under normal circumstances, for a jacks or better machine, a Royal should appear about once every 40,000 hands or so.  By altering one’s strategy it is very easy to reduce this to once every 30,000 hands or so, which is considerably more frequent.  But, it will come at a cost of lowering the payback by a significant amount too.  So, the next time you’re upset that someone else got a Royal, start worrying about how you’re playing and not what’s happening around the corner.  They may pay dearly for their Royal appearance.

           

How About That Strategy? It Works!

            This pretty much speaks for itself:

http://fbg.me/App/Posts/BlogPost.aspx?pstid=29444&pstaid=462

             Someone playing a $2 full-pay max-coin Double Double Bonus Poker hits an $8000 Royal Flush starting from a 2-Card Royal!  Without knowing who hit it (and I don’t), we really don’t know to what extent this Player follows Expert Strategy.  For all I know, he discarded a Pair of Jacks to go for the 2-Card Royal.  But, lacking this information, I’m going to use this picture to demonstrate some key components of Expert Strategy.

            Play the right machine – Well, Double Double Bonus isn’t exactly the ideal machine to play with its 98.8% payback.  However, it multiple jackpots (Royal and 4 Aces w/kicker) has made it immensely popular.  From the picture we can see that the Player chose to play one that is full-pay (1/1/3/4/6/9).  Before the payout on this hand, the Player had about $535 remaining in their bankroll.  If they play a short-pay machine, perhaps the bankroll is exhausted before they have the opportunity to win the $8000!

            Know the right strategy – This is really the crux of this column.  We don’t know what hand the Player was dealt, but we do know that he held the suited J-A.  2-Card Royals make up about 7-8% of our playable hands.  Misplaying them can be rather dangerous to your bankroll.  When my father, Lenny Frome, developed his first strategies for video poker, one of things that surprised him was that most 2-Card Royals outranked 3 High Cards.  Given the unlikely odds of hitting a Royal, intuitively, one might think that you’re better off having 3 High Cards increasing your chances of a Jacks or Better or a Straight.  But by holding 3 off-suited cards, you eliminate all chances of a Flush, reduce the likelihood of Trips and eliminate Quads and the Royal.

            ‘Unlikely’ is also a relative term.  The actual odds of hitting a Royal from a 2-Card Royal is ‘only’ 16,215 to 1.  In the grand scheme of the casino, this isn’t really all that rare.  If one out of 13 hands is a 2-Card Royal and 1 in 16,215 of these will result in a Royal Flush, then we’re talking about 1 in 210,000 hands will have this ‘fate’.  At 700 hands per hour, this means about 300 hours of play.  Depending on how serious of a Player you are, this might take weeks or months, but in a casino with hundreds of video poker machines being played 10-15 hours day each, it’s happening every day all over the place.  So, why not you?

            Of course, you increase your chances of it being you if you play your 2-Card Royals correctly.  This will, of course, depend on the specific game and paytable you are playing.  In the case of Double Double, here are some key pointers:

            -  A 10-A 2-Card Royal is NOT playable.  We hold only the Ace if we have no alternatives.

            -  We Play J-Q-K-A (off suit) over a 2-Card Royal

            -  Pay attention to all of your cards and don’t just focus on the 2-Card Royal.  You might be dealt 4-5-6 of one suit and a J-Q of another.  The 3-Card Straight Flush outranks the 2-Card Royal by a considerable margin.

            -   Do NOT discard any Pairs to go for a 2-Card Royal

            This list is hardly meant to be comprehensive.  If while reading it you realized that you didn’t know these things, you might want to brush up on your game before you spend real money.  As always, it is important to learn the strategy table of the game you intend to play and stick to it. 

            This way, maybe next time I post up a picture of a big jackpot, you can sit back and say “That’s MINE!”