Stroll Through the Strategy Table

            This week we continue our walk through a video poker strategy table.  Specifically, the strategy table for full-pay jacks or better video poker.  Last week we left off at the 4-Card Flush which was the last of the hands with an expected value of greater than 1.0.  These are the hands that result in net wins in the long run.  The rest of the strategy table have expected values below 1.0.  This means that in the long run we will not get back our entire wager.  But, that doesn't make them any less important.  When playing video poker, playing every hand correctly is critical if you want to achieve the theoretical payback.

            It could be argued that playing the hands below 1.0 correctly is more important than playing the ones above this line correctly.  First, the hands below make up the lion's share of hands.  Second, the hands below are by far more complex than the ones above.  You don't have to worry about confusing a Two-Pair with a 4-Card Straight Flush as this is an impossibility.  But a 4-Card Straight vs. a 3-Card Straight Flush might leave you shaking your head. 

            Without further ado, here are the next batch of hands on our strategy table:

·       4-Card Straight with 3 High Cards

·       Low Pair

·       4-Card Straight with 2 High Cards

·       4-Card Straight with 1 High Card

·       3-Card Inside Straight Flush with 2 High Cards

·       3-Card Straight Flush with 1 High Card

·       4-Card Straight with 0 High Cards

            The first thing you'll probably notice is that with the exception of the Low Pair, the number of High Cards is specified.  When the inventor of video poker decided to pay on Jacks or better, he added an incredible layer of complexity to the strategy.  Simply put, in any hand without a Pair or better, any card that is a Jack or higher is worth considerably more than any other card.  The reason should be fairly obvious.  We have the opportunity to win with High Pairs.  For each High Card in the hand, we have three additional cards that we can draw that will turn our hand into a winner.  These three cards add just over 0.06 to the expected value of the hand. 

            Sometimes, this 0.06 means nothing and sometimes it means everything.  We just need to look at the first three hands to see the impact.  If you have a 4-Card Straight with 3 High Cards and a Low Pair, you keep the partial Straight.  If you have a 4-Card Straight with 2 High Cards and a Low Pair, you keep the Lower Pair.  These hands are not very common, but they illustrate the impact of the High Card.   So, if you have 10-10-J-Q-K (assuming no 3-Card Royal), then you keep the Straight.  If you have 9-10-10-J-Q then you keep the Low Pair.  Having a 9 instead of a King lowers the expected value so that it falls just below that of the Low Pair.

            If we keep moving down the chart, we find that the next entry is the 4-Card Straight with 1 High Card.  As this is adjacent to the 2 High Card version, there really is no impact in this case.  We could lump these two hands together if we want to remove 1 hand from the strategy table.  We keep them separate because there are versions of video poker where it is relevant and we want to make sure the Player doesn't get 'lazy'. 

            In between a 4-Card Straight with 1 High Card and a 4-Card Straight with 0 High Cards we find 2 other hands.  They are both variants of a 3-Card Straight Flush.  The first is an Inside Straight Flush with 2 High Cards and the second is a 3-Card Straight Flush with 1 High Card.  This can start looking more confusing than it really is.  Most of these hands CANNOT occur in a single hand.  It is not possible to have both a 3-Card Straight Flush with 1 High Card and a 3-Card Inside Straight Flush with 2 High Cards.   But, you can have a 4-Card Straight with 1 High Card with a 3-Card Straight Flush with 1 High Card.  (8C 9D 10D JD 4H).  We learn from the strategy table that the right play is the 4-Card Straight.

           

            One last point that I should mention.  All the 4-Card Straights to this point have been Open Straight Flushes - meaning that they can be completed on both ends.   This means that as we have completed about 60% of the strategy table, we have accounted for all Pairs and for all 4-Card Straights (Open ended) and 4-Card Flushes.  The remaining 40% of the strategy tables contains very 'not pretty' hands.  It is a mish-mosh of 3-Card Straight Flushes, Inside Straight Flushes and even Double Inside Straight Flushes, along with 2-Card Royals and hands with just High Cards.  To make matters worse, these 40% of the entries account for nearly 50% of the hands.  

            Next week, we'll continue our stroll through the strategy table.

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.

           

Quads are the Key

For those of you who read my column regularly, you are probably now well aware that a full-pay jacks or better video poker machine pays 99.5%.  Many people are still confused, however as to what this means.  It does NOT mean that if I start with $100 I will walk away with 99.5% or $99.50.  It means that in the long run, you could take the total amount you wager (NOT your bankroll) and multiply it by 0.5% (the 'loss' rate or 100% minus the payback) and this should be the amount you have lost over time.  So, if you play 10,000 hands over the course of a year (or a month or a decade) and you play max-coin $1 machines, you would have wagered $50,000 and can expect to lose about $250. 

            In video poker, however, 10,000 hands isn't really the long run.  Don't get me wrong, it is certainly approaching the long run.  But, given that a Royal Flush should occur about every 40,000+ hands, it would be hard to declare 10,000 hands to be the long run.  If you've hit at least one Royal, you would be ahead of the game.  If you haven't, it would be totally fair to say you are behind because you still have 30,000+ hands to go.  Royal Flushes account for about 2% of our payback.  So, if you were to NEVER hit one, you'd theoretically be playing only a 97.5% game.

            With a hand frequency of 1 in 40,000+, Royals sort of march to their own drummer.  You might hit 2 or 3 in 40,000 hands or you might go 100,000 hands without hitting one.  When you hit one, you're going to have a very good month and when you don't, well, it will be harder to even break even.

            Four of a Kinds, on the other hand, should occur about 1 in about 420 hands.  With the average Player playing hundreds of hands per hour and perhaps thousands in a session, this hand becomes critical to our chances of success over a session or two.  It accounts for 6% of our overall payback.  Relative to the other hands, this is not necessarily large, but it is a hand that is frequent enough that you expect to hit it over a session, but not frequent enough to be sure you'll hit your fair share over a few nights.   If you were to play 10,000 hands, you'd probably find that the frequency of High Pairs and Trips and Straights are very close to what they should be.  Royals will by very definition have to be either more frequent or less frequent than expected, but Quads can be just about anywhere over that period of time. 

            In theory, you should hit about 24 of them over that time.  The math says you very likely could hit only 12 or as many as 36.  If you hit 12, you're about 3% short in payback.  Assuming you haven't hit a Royal and you're now 5% short.  The odds of coming up a winner over that span is very unlikely as you'll be playing at 94.5% and hoping the other hands come up big - which simply isn't very likely.

            Conversely, if you've hit36 of them, you'll be at 100.5% EVEN if you haven't hit a Royal.  A winning session is not guaranteed but certainly more likely.  Over time, the frequency of the Quads will slowly head towards that 24 per 10,000 hands, but your results in the short or medium run is heavily dependent on hitting your four of a kinds.

            As with anything video poker, the number of Four of a Kinds you get is at least partially attributed to luck.  We've all played for hours and been dealt dozens of Three of a Kinds to watch NONE of them turn into Quads.  We've also all sat there and drawn 3 Kings to a single King.  Nothing that happens is truly out of the ordinary.  However, you can increase your chances by playing the right strategy.  If you hold a 4-Card Straight OVER a Low Pair, you are going to greatly reduce your chances of Quads.  If you hold 3 High Cards instead of just the 2 that are suited, you will lower your chances for getting Quads.  The reason why we hold only the 2 suited cards is both to give us a chance to hit the Royal AND to increase the chances of Quads.  Both of these hands are reduced to ZERO CHANCE if you hold 3 off-suit High Cards!

            Of course, if you choose to hold a Low Pair OVER a 4-Card Flush you may increase the frequencies of Four of a Kinds, but you'll do so at your own peril.  Quads are important, but not so important that you should be throwing the proper strategy out the window.

Rare Gems - Straight Flushes

            One of the ironies about video poker paytables is that they don't always reward hands more for being more rare.  If I were to ask you which occurs more often in video poker - a Flush, a Straight or a Full House, I'm guessing most of you would say a Straight, followed by a Flush and lastly a Full House.  It is really a trick question.  Without knowing what the paytable is, there is no way to answer the question accurately.  The only thing we know is that, in general, a Full House outranks a Flush, which outranks a Straight. 

            On a full-pay video poker machine, assuming you use Expert Strategy, you will actually hit more Full Houses than either of the other two.  A Straight will occur just slightly more often than a Flush.  Upon close inspection, we realize that this is by far a product of the payouts for each hand than it is a product of the hands themselves.   If we take a look at the game of All American Video Poker - which would appear to now be obsolete - we will see a very different pattern develop.  In All American, a Straight, Flush and Full House all pay 8.  With no reason to go for one or the others, the pure probabilities of hitting each hand begin to show up.  As a result, the frequency of Straights and Flushes increase dramatically, to the point where they occur nearly twice as often as a Full House.

            A similar phenomenon occurs with a Straight Flush.  Generally speaking, it occurs just about 4 times as frequently as a Royal Flush, while paying only 1/16th of the amount.  Or we can look at it the other way and say that it is more than 20 times as rare as a Four of a Kind while only paying twice as much.  When we throw in the Bonus Video Pokers, it only looks worse.  This far more rare hand might actually pay LESS than many of the Quads we can hit, which are far more common.

            Of course, I'm wondering how many of you have hit nearly as many Royal Flushes as you've hit Straight Flushes.  I doubt you remember your Straight Flushes as vividly.  Winning $62.50 on a max-coin quarter machine isn't quite as memorable as a cool $1000, but that isn't my point.  If you use Expert Strategy on a jacks or better machine, you should hit a Royal every 40,400 hands or so and a Straight Flush every 9200 hands.  The key phrase is "if you use Expert Strategy."  Since most Players, at best, use pieces of strategy, I'm guessing that the Straight Flush shows up far less often because the partial Straight Flush is frequently overlooked when the Play.

            If dealt the following, what's the right play?

J♠        8♦        Q♦       3♥        9♦

            Do you play the 4-Card Inside Straight with 2 High Cards, the 3-Card Double Inside Straight with 1 High Card or the 2 High Cards?  As always, there is just one way to determine the right play.  We go to the Expected Values of each.

            Calculating the Expected Value for the 4-Card Inside Straight is fairly easy.  We can draw the Straight with 4 cards and we can draw a High Pair with 6 more.  This will return 22 units to us.  Divide by 47 and we get a result of just below 0.47.  For the other two, I ran them through a program I have that calculates the exact Expected Value given the specific discards.   The Two High Cards have an Expected Value of just below 0.50 and the 3-Card Double Inside Straight Flush has an Expected Value of just below 0.53.  This is the proper play. 

            While the odds of hitting the Straight Flush are 1 in 1081, this is still far greater than hitting it with either of the other two hands (it is zero in these cases).  Ironically, it is not the tremendous payout of the Straight Flush that causes us to play the hand this way.  By holding a 3-Card Straight Flush, we give ourselves numerous chances to hit just Straights and Flushes - a combined 1 in 20 (roughly).  Throw in opportunities for Three of a Kind and Two Pairs and this hand simply beats the others.

            Now, no one expects you to calculate the Expected Value of even the 4-Card Inside Straight on the fly or to carry a small computer to run my program that calculates the exact Expected Value for each hand.  It is much easier to simply use a strategy table that lists out each playable hand.   If we look up the three hands in a strategy table, we find a 3-Card Double Inside Straight Flush has an Expected Value of 0.54, the Two High Cards have an Expected Value of 0.49 and the 4-Card Inside Straight with 2 High Cards doesn't even make it onto our strategy table because the Two High Cards always outranks it.  These values are the average of all hands of that type so they don't always equal the exact Expected Value taken into account the exact discards.

            In the end, the frequency of a hand occurring is a product of the paytable and following the right strategy.  If you want to get your share of Straight Flushes, you can't do a lot about the former, but the latter is fully in your control.

Is the Search for Perfection Overrated?

            

            One of the traits I inherited from my father, Lenny Frome, is that I am a perfectionist.  This is not to say that I am perfect (far from it).  The joke in our family was always that one of my sister's brought home a '99' on a test and my father, not missing a beat said "why not a 100?"  Fortunately, for me, by the time I came along, he learned to temper his ways a bit.  I try to focus most of my perfectionism inward.  Nobody takes it harder when I find a mistake in my work more than I do.

            At the same time, I try very hard to be practical about things too, where strategy is concerned.  Us mere mortals do have limitations to our ability to memorize dozens of video poker game strategies.  It is why I strongly recommend that you learn one or two different games and do your very best to 'perfect' the strategies to those games.  But, what are you to do if the games whose strategy you memorized are not available when you go to the casino or if you're just in the mood to try something different?

            In these cases, you just have to use some common sense.  If you try to bring your Jacks or Better strategy over to Joker Poker, you may find yourself in deep trouble.  But, what happens if you use your basic full-pay Jacks or Better strategy on a full-pay Bonus Poker machine?  What will this really cost you in theory?  Calculating this - with the help of some of my video poker analysis programs - is relatively easy.

            We simply run the numbers on a full-pay jacks or better machine using Expert Strategy for jacks or better.  From this program, we extract the frequencies of all of our winning hands.  We then use these frequencies against the Bonus Poker paytable to get a theoretical payback of Bonus Poker using jacks or better strategy.  We then run the numbers on Bonus Poker using Expert Strategy for Bonus Poker and compare the results.

            The theoretical payback for Bonus Poker using Expert Strategy for Bonus Poker is 99.16%.  The theoretical payback for Bonus Poker using Expert Strategy for jacks or better is 99.15%.  If you were to play all 2,598,960 possible 5-card deals using jacks or better strategy you would find that you cost yourself about 200 coins.

            In other words, while it is still preferable to learn the right strategy for Bonus Poker, if you use your jacks or better strategy, your bankroll will not take a big hit.  However, this should not give you free license to play jacks or better strategy on any game you want.  If you were take your jacks or better strategy to a DOUBLE Bonus Poker game, you would find that it will play at about 99.6%.  This might sound good (after all jacks or better itself plays at a bit less than this), but you have to remember that Double Bonus Poker is one of the few positive games out there.  If you play it using the proper strategy, it can afford you a 100.1% payback.

            So, what's the point?  Good question.  Nobody should expect perfection when they head to the casino.  The simple mathematical fact is that every deviation from perfection, however, will cost you.  It might be peanuts as in the case of playing Bonus Poker using jacks or better strategy.  If you play a hundred hours a year as a $1 max-coin Player, you'll cost yourself $30 per year.  We'd all rather have that $30, but we're not talking a significant amount of money.  Do the same on our Double Bonus scenario and we're talking about $1500 per year, which I dare say is QUITE significant.

            So, while we shouldn't shy away from video poker because we might make a few mistakes, we should still be prepared to learn the right strategy for each game and to do our very best to utilize it every time we play.  This is the very essence of Expert Strategy - Know which games to play, know what strategy to use and know what to expect.

            For those of you who want to learn the subtle differences between jacks or better strategy and Bonus Poker strategy, both strategies can be found in Expert Video Poker for Las Vegas and Winning Strategies for VideoPoker.