Get the Inside Scoop

            Last week I began reviewing the strategy table for full-pay jacks or better video poker.  I got about 20% of the way through the table by volume, but not very far in terms of useful information.  The top 8 hands were mostly of the no-brainer category as they were the pat hands with the exception of the 4-Card Royal.  This week, I'll keep moving down the table and provide some insight into the nuances of video poker strategy.  Please remember that this particular strategy is applicable ONLY to full-pay jacks or better.

            After a Straight, we find the following entries on our table:

·       4-Card Straight Flush

·       Two Pair

·       4-Card Inside Straight Flush

·       High Pair

·       3-Card Royal Flush

·       4-Card Flush

            The first thing you might notice about the above entries is that we have two entries for a 4-Card Straight Flush and a 4-Card Inside Straight Flush.  There is a big difference between the expected values for Straights that are open and those that are Inside (or Double Inside).  The common definition of Inside Straight is when the opening is in the middle and not on the ends (i.e. 5-6-7-9).  However, this leaves off some Inside Straights.  It is more accurate to define a 4-Card  Inside Straight as one that can only be filled ONE WAY.  So, an A-2-3-4 can only be filled with a 5 and thus is an Inside Straight.   With this definition you can see that an Inside Straight can be completed with only 4 cards while a regular Straight can be completed with 8 cards.  Straight Flushes are no different - except they have the possibility of being turned into Flushes as well. 

            In this particular case, there is really no benefit to splitting out the 4-Card Straight Flushes.  The one hand that lies between them can't possibly be a 4-Card Straight Flush (Inside or not).  We show them separately because in some version of video poker, the hands that appear in between may be able to overlap with them and we will find that in some cases we will want to keep a 4-Card Straight Flush ONLY if it is not an Inside Straight Flush.  Also, as we will see as we move down the table, this distinction becomes very important as we take a closer look at 4-Card Straights.

            The 4th entry on the table is a critical one - High Pair.  It is the 4th most common hand.  Thus, playing it correctly is very important.  Looking at the entries above it and below it what we learn is that a High Pair is played OVER any 4-Card Straights and 4-Card Flushes.  We will, however, play all 4-Card Straight Flushes over a High Pair.  But, we will NOT play a 3-Card Royal over the High Pair.  So, if you have a suited J-Q-K along with another Queen, you stick with the sure winner - the Pair of Queens.

            Below High Pair, we have a 3-Card Royal Flush and a 4-Card Flush.  There is much to learn here as well.  The most obvious is that if you have a 3-Card Royal and a 4-Card Flush, we hold the 3-Card Royal.  This can be a tough choice because the likelihood of hitting the Royal is still relatively small.  But, by holding a 3-Card Royal we give ourselves more chances for a Straight.  We might still hit a Flush and we have the longshot at the Royal.  Also, with a 3-Card Royal, we leave ourselves 2-3 cards that can be matched up for a High Pair.  The expected values are not really all that close with a 1.41 for the 3-Card Royal and 1.22 for the 4-Card Flush.  The decision is relatively clear. 

            From these entries we also learn that if the Player has a 3-Card Royal that is also a 4-Card Straight Flush (8-10-J-Q), we hold the 4-Card Straight Flush.  With the 4-Card Straight Flush, we still have many chances for Straights and Flushes so we don't throw away the extra card even if it gives us a chance to get the Royal.

            I've stopped at this particular point in the Strategy Table because the 14 hands I've listed (over the past 2 weeks) are the only ones with an expected value greater than 1.0.  That means these hands are net winners in the long run.  Some will be winners 100% of the time.  Some will not.  But in the long run, we can expect to get more back than we wagered.  These hands make up about 40% of the table and about 25% of the total hands dealt.  Beginning next week, we'll review the hands with an expected value below 1.0.  Even though these are losers in the long run, it doesn't make them less important.  In fact, they may be more important because they account for a larger percentage of hands dealt.

           

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.

It's Mathemagic!

          A couple of weeks ago, a colleague discussed with me the possibility of creating a simple three card sidebet for a game he was working on.  Essentially, it was no different than the Pair Plus wager for Three Card Poker. As a favor, I quickly gave him the frequencies for the hands and made sure there was a note that said that these applied ONLY to a standard 52-card deck.  The next day I received a frantic voice mail that stated that he wanted these numbers for a blackjack-style shoe and what did I mean that the numbers only applied to a single deck.  Since there is still the same number of cards of each type relative to the others, shouldn’t the frequencies stay the same?

          This led to a lengthy conversation in which I tried to explain why the frequencies change as the number of decks goes up.  Generally speaking, the frequency of sets (i.e. Pairs, Trips, etc…) goes up while the frequency of runs (Straights) goes down.  Below is a table that shows the frequencies of the winning hands in Pair Plus for both a single deck and a 6-deck shoe.

Hand	1-deck	6-decks
Mini Royal	0.0181%	0.0172%
Straight Flush	0.1991%	0.1896%
Three of a Kind	0.2353%	0.5248%
Straight	3.2579%	3.1021%
Flush	4.9593%	5.8424%
Pair	16.9412%	19.4918%

          The table clearly shows that which I just stated.  The frequency of a Three of a Kind more than doubles when we move to 6 decks.  A pair occurs about 15% more often.  Flushes occur about 20% more often.  In the meantime, Straights decrease by a mere 5%.  Our Royals and Straight Flushes decline slightly as well as they are a blend of Straights and Flushes.  The overall hit frequency goes up about 15% from 25.6% to 29.2%.  Is it any wonder that a paytable that works well for one deck would be a disaster (for the casino) with six decks?   If we were to take the standard Pair Plus paytable and apply it to a 6-deck version, the payback would go from just under 98% to nearly 115%!

          My colleague was having a tough time believing me, and I did my very best to try and explain it.  I used the Three of a Kind as an example.  Once the Player is dealt a Pair, he is looking for 1 of the 2 remaining matching cards with 50 remaining in the deck for a single deck game.  Thus he has a 4% chance of drawing the Three of a Kind at that point (2 out of 50).  When dealing from a 6-deck shoe, there are still 22 cards of that same rank left in the shoe out of 310 cards.  This gives the Player a better than 7% chance of drawing a Three of a Kind from a Pair.  With a single deck, the probability of drawing additional cards of the same rank decreases greatly as each card of that rank is dealt out.  Whereas, with a large shoe, the probability of drawing cards of the same rank does not decrease nearly as much as each was in used up.

          Taking this to the extreme, imagine a game where 4 cards are dealt and you are trying to draw a Four of a Kind.  In a single deck game there is are only 13 ways to do this.  By the time you have been dealt Three of a Kind, you have only a 1 in 49 chance of being dealt the fourth card of the same rank.  In a 6-deck version, you would a 21 in 309 chance – which gives you a 3 times greater chance of hitting it from that point.  In total, you would have 13/270,725 (0.0048%) probability of getting a Four of a Kind from a single deck vs. a 0.0357% probability from 6 decks.  This translates to nearly 7 ½ times as likely to get it using six decks.

          There are many lessons that can be learned from all this.  The first is a simplistic lesson in why casinos like large shoes for games like blackjack.  It is much harder for a Player to count cards and trying to figure out what is coming up.  Just because a few small cards have been dealt doesn’t mean large cards are that much more likely than normal to show.  With a single deck, just a few small cards means the probability of large cards increases by a far larger margin.

          The second lesson is that it is not always that easy to compare paytables from one game to another.  If a casino were to offer a Pair Plus type wager from a large shoe, they would have no choice but to offer lower payouts than they do for a single-deck game.  However, that would not necessarily mean that the overall payback is lower.  You need to make sure that you are always compare apples to apples.  A game dealing 7 cards can’t offer the same payouts as one dealing 6 for like poker hands, but you can’t just look at the payouts and say that one is ‘worse’ than the other.  You have to look at the actual frequencies of each winning hand and determine the true overall payback.