Wednesday, December 28, 2011

The Details Behind the Advice

I’ve spent the last couple of weeks trying to get the beginners among you to make a relatively simple adjustment to your strategy. It involves four relatively common hands – high pair, 4-card flush, low pair and 4-card straight.
As I explained last week, they are played in this order because of their expected values. This week, I will walk through the calculation of the expected values for each of these hands.
We start with the easy one first. It is easy because EVERY high pair has exactly the same Expected Value (EV). Since we already have a pair of jacks or better, we don’t have to worry about what are the specific cards discarded as they cannot help the hand nor interfere with other hands being formed.
When dealt a high pair, we will draw three cards. There are 16,215 combinations we can then draw from the remaining 47 cards in the deck (47 choose 3). Let’s look at the results of all of these draws:
45 will result in a four of a kind paying 25 each for a total of 1,125.
165 will result in a full house paying nine each for a total of 1,485.
1,854 will result in a three of a kind paying three each for a total of 5,562.
2,592 will result in a two pair paying two each for a total of 5,184.
11,559 will result in a high pair paying one each for a total of: 11,559.
The Grand Total is 24,915.
We divide the grand total by the number of combinations to arrive at the Expected Value of 1.5365. Every high pair has this exact EV. By itself, this number means relatively little in terms of our strategy.
Yes, it does tell us that we can expect to win about 1.5 units back when we have a high pair, on average, but it doesn’t tell us if we should play a 4-card flush or a high pair when we have both.
This will generate very similar results to our high pair. The only (and very BIG) difference is that all of those high pair hands at the end will now end up as low pairs and pay nothing. Thus, we will have a grand total of only 13,356, which when divided by 16,215 gives us an Expected Value of 0.8237.
The 4-card flush and the 4-card straight each have 47 possible draws. The flush can result in nine flushes paying six each – for a total of 54.
The straight (NOT INSIDE) can result in eight possible straights paying four each for a total of 32. However, depending on how many high cards each has, it may be possible to wind up with a high pair as well.
For each high card that is in the 4-card flush or 4-card straight, three additional hands can wind up as a high pair instead of a losing hand. These additional three units when divided by 47 possible combinations means that each high card adds about 0.0638 to the Expected Value of our 4-card flush or 4-card straight.
So, a 4-card flush with zero high cards has an expected value of 1.15 (54 divided by 47). If there is one high card, we add .064 to this to get to about 1.21. With two high cards it climbs to about 1.28.
With three high cards – well, we would have a 3-card royal and that’s a whole different hand! So, a 4-card flush has an EV of somewhere between 1.15 and 1.28.
Since no other hand has an EV in between these two, we don’t bother separating these hands out on our strategy chart. Instead, we take the average of ALL 4-card flushes and say that its Expected Value is 1.22.
With regard to a 4-card straight, the Expected Value with zero high cards is a paltry 0.68. With one high card it goes up to 0.74. With two high cards it goes 0.81 and with three high cards to 0.87. Technically, a 4-card straight with 4-high cards is an inside straight (only one way to complete it) so its EV is much lower.
Because numerous other hands, including our low pair have an Expected Value in this same range, our strategy table shows each of these hands separated out.
So, when we look at all of these hands and rank them from high to low in terms of their Expected Values, we come up with the following:
High Pair: 1.54
4-Card Flush: 1.22
4-Card Straight with three high cards: 0.87
Low Pair: 0.82
4-Card Straight with two high cards: 0.81
4-Card Straight with one high card: 0.74
4-Card Straight with zero high cards: 0.68
It is based on these Expected Values that our strategy is derived. I’d like to raise two final important points. First, note that the 4-card straight with three high cards actually outranks the low pair – which is in conflict with the simple rule I gave two weeks ago.
While you should play this 4-card straight OVER the low pair, this particular combination is so rare that ignoring it while you work on learning the strategy will not cost you much. The ONLY way this hand can occur is 10-10-J-Q-K.
This leads to the second important point. For the purposes of this part of the strategy, ALL of our 4-card straights are outside – meaning they can be completed on either end. The other type of straight is an "inside," which has a gap in the middle or has an ace on one end or the other.
These can be completed only one way and have a much lower Expected Value. In Jacks or Better, most inside straights are not even playable.
I’d like to take this opportunity to wish everyone a Happy and healthy New Year and remind everyone to make their resolution to break the slot habit in 2012!

Wednesday, December 21, 2011

Clarity on a Little Advice

Last week’s column gave some simplistic advice to beginners who are not yet ready to sit down and really learn the strategy for video poker. It discussed the relative rankings of four of the most common hands – high pair, four-card flush, low pair and four-card straight.
While I gave the expected values for each of these hands, along with some explanations as to why the rankings are what they are, this week I want to stress that these explanations are not the critical part of the process.
The strategy for video poker is based on one thing – math.
We don’t keep a high pair over a 4-card flush because the high pair is a sure winner. If this were the case, we’d keep a high pair over a four-card straight flush, too (but we don’t!). The fact that the high pair is a sure winner explains why its expected value is as strong as it is, but it is the actual value of this expected value that puts the high pair where it does.
So what is this "expected value" I keep talking about?
It is the average amount of coins we expect to win over the long run from that hand.
How is it calculated?
It is calculated by looking at EVERY possible draw given the 5-cards already dealt.
Say what?
There are 2,598,960 ways to deal five cards from a 52-card deck. For each of these ways, there are 32 different ways to play each – ranging from discarding all the cards to discarding none of them. For each of these 32 ways to play a hand, there is a varying number of possible draws.
If we discard one card, then there are 47 possible draws (each of the 47 remaining cards). If we discard three cards, then there are 16,215 possible draws (choosing three cards from 47). A computer program goes through every possible draw and tallies up the winning hands for each of the 32 ways to play a hand.
It then computes the average number of coins returned for that way. This is the expected value for that particular way of drawing. It compares the expected values for each of the 32 ways and whichever has the highest one is the proper play for that deal and is deemed the expected value for that deal.
An example usually helps to shed some light on this process. Assume you are dealt: 4 of clubs, 5 of hearts, 5 of clubs, 5 of spades, 7 of diamonds.
We recognize the three-of-a-kind (5’s), the EV of which is calculated as follows:
Drawing two cards from the 47 remaining in the deck will create 46 four-of-a-kind winners (a five combined with each of 46 remaining cards). Sixty-six draws will end as full houses (six pairs in all ranks but 4, 5, and 7; 3 pairs of 4 and 7) while the remaining 969 draws do not improve the hand but instead leave it as a three-of-a-kind.
In summary we have:
46 4-of-a-Kind paying 25 each,
66 Full Houses paying 9 each,
969 3-of-a-Kind paying 3 each,
We calculate the total payout as 4,651, which is an average of 4.30 for each of the 1,081 possible draws. Therefore, the expected value of this deal/draw combination is 4.30.
As should be fairly obvious, if we try to play this hand in any of the other 31 ways, the expected value will NOT be any higher than 4.30 and thus this is also the expected value of this deal.
As all three-of-a-kinds have exactly the same expected value, this is ALSO the expected value of all. We will find this value on our strategy table.
Next week, I’ll walk through the four hands (high pair, low pair, four-card flush and four-card straight) I used in last week’s column. This will explain why the strategy I described last week doesn’t just make some sort of logical sense but is the right play mathematically.
I’d like to take this opportunity to wish everyone a happy holiday and a very happy and healthy 2012!

Tuesday, December 13, 2011

A Little Advice

            Last week's column was a gambling related philosophical debate about perfect vs. good enough.  This week, I'm going to the other end of the spectrum.  It is nearly impossible to define a 'bad' strategy as there really is no end to how bad a Player can play most games.  Playing every hand in Three Card Poker would probably meet the definition of a bad strategy, but is it worse than Folding every hand below a Pair?  Probably not, and I'm not going to waste my time to try to find out.

            This is not to say that every strategy that isn't perfect or as per last week's column 'good enough' would necessarily meet the definition of 'bad'.  I don't consider playing Three Card Poker with the strategy of Play any hand with a Queen to be good enough, but I can't really call it a bad strategy either.  With a game like Three Card Poker, there isn't really much to learn so you draw your line in the sand where you do and that's how you play it.

            A game like video poker is far different.  For anyone that doesn't use Expert Strategy, you might be hard pressed to find two people who used identical strategies.  In reality, they may be TRYING to use Expert Strategy (or some other particular strategy) but due to its complexity, they make a variety of errors along the way.  Then there are the multitudes of Players who just play by the seat of their pants, pretty much oblivious to the math that should be guiding them.  To these Players, getting them to even good enough will be quite a challenge.

            But, no matter what level they play at, if they just learn a few simple strategy points that might help them get a little closer to Expert Strategy then at least it is a step in the right direction.  So, today's column is for these Players.  I would like you all to consider learning just this small part of the strategy and trying to implement it.  You may still be a long ways away from playing Expertly, but hopefully, we can save you just a few bucks along the way and add to your enjoyment too.

            Here goes:
            1)  High Pair
            2)  4-Card Flush
            3)  Low Pair
            4)  4-Card Straight

            This strategy only means something on the hands that are either a 4-Card Straight or a 4-Card Flush and are also a Pair.  Approximately 25% of all 4-Card Straights and Flushes fall into this category, so these hands are fairly common.  This is why it is imperative that these hands be played correctly.  Let's take a closer look at why you should play the hands as described above and learn how these are NOT close calls.

            The High Pair is the only sure winner in the bunch, but this is NOT the reason it is at the top of the chart.  The determining factor is always the expected value of the hand, which is the average amount we expect to win with that hand over the long run.  Sometimes, the sure winner is not the right answer, but in this case it is.  The expected value of our High Pair is 1.54 which reflects the opportunities to turn this into Two Pair, Trips, Full House and Quads. 

            Next up is the 4-Card Flush which will win for us in the long run.  This is NOT to say that we will have more winning hands than losing hands.  With 9 opportunities to complete a Flush and perhaps a few more to complete a High Pair (depending on the exact makeup of the 4-Card Flush), we can expect to win with this hand only 20-30% of the time.  But since many of these will win with a Flush, the wins will be significant.  The expected value of a 4-Card Flush is 1.22.  It will be a smidge higher if you have 1 or 2 High cards and a bit lower if you have none.  If you have 3 High Cards, you have a 3-Card Royal and that takes precedence over the 4-Card Flush, but not the High Pair.

            While the Low Pair has the exact same probabilities as the High Pair of winding up as Two Pair, Trips, Full House or Quads, the fact that it starts as a losing hand is enough to bring its expected value all the way down to 0.82.  That means in the long run, this is a losing hand.  It is the second strongest losing hand (behind the relatively rare 10-J-Q-K Straight, which is also the ONLY exception to the rule I'm presenting here as you hold this 4-Card Straight over a Low Pair, which can only happen with a Pair of 10's).  The Low Pair is also BY FAR the most common hand in video poker, accounting for nearly 30% of all hands.

            Lastly, we have the 4-Card Straights.  While a 4-Card Straight with 2 High Cards ranks only slightly below the Low Pair with an expected value 0.81, it is still below it.  It only gets worse with 4-Card Straights with 1 High Card or 0 High Cards with expected value of 0.74 and 0.68, respectively.  These may not seem like big differences, but they will eat at your bankroll over time.

            It would still be far better for anyone reading this to become a truly Expert Player, but any improvements in your strategy are still better than none.  To help you on your way, we continue with our holiday special.  We are offering Winning Strategies for Video Poker, Video Poker: America's National Game of Chance and Expert Video Poker for Las Vegas for $5 each, which includes postage and handling.  Feel free to order as many as you'd like as they make great stocking stuffers!  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.  We'll do our best to get them to you before the holidays.

Tuesday, December 6, 2011

Perfection is the Enemy of Good Enough

            Recently, while my teenage son and I were debating something, he responded with "perfection is the enemy of good enough."  My initial response was to shoot back "good enough is the enemy of perfection."  Since this highly philosophical discussion, I've given both of these phrases a lot of thought.

            I'm very well aware that I am a perfectionist who was raised by a perfectionist.  If you brought home a 99 on a test, my father wanted to know why you didn't get a 100.  If there is such a thing, however, as a realistic perfectionist, I think both by dad and I would qualify.  We strive for perfection, but also realize that it is often not realistic to truly attain it all the time.  I think this is why I found the aforementioned quotes to be both interesting and a little befuddling.

            My initial reaction that good enough is the enemy of perfection goes to my basic notion that we should always strive to be perfect.  Over the years, I've been asked many times regarding the strategy for Three Card Poker and if it really matters if you go with Q-6-4 or just Q-High.  The impact to payback is barely noticeable.  You might play for hours before getting a hand that Plays under one strategy but not the other.  Yet, the notion of settling for the easier Q-High frustrates me so.  Clearly the strategy is 'good enough.'  But, is remembering Q-6-4 SO hard that you one needs to go with Q-High?  To me, this is a case where good enough became the enemy of perfection.

            There were times my father's work on video poker was criticized (mildly) by other analysts for being less than perfect.  On one hand, my father was not prone to doing things less than perfectly - especially math work.  On the other hand, he taught himself how to program a computer at age 60, so this was not totally his comfort zone.  In a nod to that realistic perfectionism I mentioned earlier, my father's strategies for video poker were not designed to be 100% perfect.  They were designed to be played by humans.  And, not a bunch of rocket scientists, but the masses.

            The process that my father used to analyze video poker was rather similar to the same one I use, which is most likely not all that different from the ones created by anyone else.  We all have different degrees of shortcuts we use to speed up the process but the basic idea is the same.  We look at each of the 2,598,960 possible initial 5-card deals from a 52-card deck.  We then analyze each of the 32 possible ways to discard and review each of the myriad ways to draw to each of these 32.  Whichever of these 32 ways results in the highest expected value is the proper way to play the hand. 

            The calculation to do the above is absolute and assuming no error in the process will be 100% accurate.  In other words, it will be PERFECT.  So, in a perfect world, a Player could sit down at a video poker machine, press the Deal button and then enter the five cards he was dealt into an APP on his phone, which would run the process I just mentioned and tell him exactly which cards to discard.

            Unfortunately, the casinos are not too keen on this idea.  In fact, I was recently sitting at a Blackjack table and pulled out my phone to check e-mails while the Dealer was shuffling and got reprimanded.  I knew you couldn't use such devices at the table, but I assumed this meant while the game was in progress, not while waiting for the shuffle!  So, sitting at a video poker machine with your tablet in your hand will probably not be allowed.

            Because of this, the next best thing is that the results of analyzing all of these hands need to be summarized a bit.  This is what we call a strategy table that lists the rankings of all the hands in order of their expected value.  Certain hands become essentially 'exceptions to rules' when we try to summarize the hands.  These exceptions could be listed as their own rows on the strategy table, but what would the impact be if the strategy table grew to be 50 or 60 rows instead of the usual 35 or so?   By ignoring these exceptions we cost ourselves MAYBE 0.01% or 0.02% of payback, but we greatly simplify the strategy table, thus reducing the probability of errors.

            In this case, my son was right as perfection could be the enemy of good enough.  My father could have put together a perfect strategy table, but if learning it became that much harder so that the likelihood of errors increased to the point where an average person would lose more in errors than he would gain in playing 'perfectly' - would this still really be 'perfect'?

            At the end of the debate, it would appear that my father had already resolved the issue for us - and we were both right!

            As we are approaching the holiday season, Gambatria would like to offer to all of our readers a deal that may not be perfect, but is certainly better than good enough.  We are offering Winning Strategies for Video Poker, Video Poker: America's National Game of Chance and Expert Video Poker for Las Vegas for $5 each, which includes postage and handling.  Feel free to order as many as you'd like as they make great stocking stuffers!  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.  They'll ship 1st class mail (or priority mail in some cases) so you can get them in time for the holidays.

Friday, December 2, 2011

Vintage Lenny Frome - Video Poker is NOT Slots!

This article was first published in about 1992 by my father Lenny Frome.  Keep that in mind as you read through some of his comments and realize just how much has changed in the nearly 20 years since!

Video Poker is NOT Slots!
by Lenny Frome

            Every time we write a column for a new publication, we do so with a great deal of uneasy feeling.  After all, the readers who pick up this journal after a session at the poker tables or in the Bingo parlors look at Video Poker players with disdain.  No matter how special we consider our machines, they look at them as "just slots".

            In 1988 Las Vegas had a poker room paper called of all things, POKER ROOM. Within days of accepting our very first Video Poker article, the publication closed its doors. Imagine our guilt feelings as we contemplated that just planning to put Video Poker into print could cause a gambling paper to close. Maybe they were "just slots" then.

            In the four short years since , Video Poker has come of age.  From just a handful of game versions, there are at least 50 unique versions, which with their various pay-tables, create literally hundreds of different games.  Today, the term "Video Poker" doesn't hardly give a clue as to what kind of game we're referring to.

            The public by and large has learned to respect this family of games for several reasons. Most analysts attribute its popularity to the man-machine interaction--the decision making by the player which affects the outcome.  Others claim the players enjoy their privacy and are never intimidated.  Those reasons don't satisfy me because for a long time Video Poker languished in Las Vegas.  When the machines paid on on two-pair or better, they were a drug on the market.  Nobody knew how to play them and even when they did approach expert play, the payback of 90% disenchanted the public.

            When the pay-table was revised to pay on Jacks or Better, the public flocked to them.  Nobody, including the casinos really could explain this phenomenon because it took quite a while before the 99.6% payback on expert play was proven.  Meanwhile, the public could sense that they won much more often and played longer.  In the long run, players still left money in the machines but they enjoyed the time on them.  Today, one-third of casino revenue is derived from Video Poker.

            Outside of Las Vegas the payback is necessary lower which makes it even more important for players to learn how to play correctly.  To become a good player is easy once becomes be aware of several key factors:


(A)  The game is governed purely by known mathematical probability;  if you don't believe that, you cannot become a good player.

(B)  Once the deck is defined and a pay-schedule displayed, the optimum strategy for hold/discards on every hand is known, along with the payback percentage and the average number of each level of winners.

(C)  Unlike reel-slots, which can have their payback altered almost at the whim of the casino with absolutely no warning to the players, Video Poker payback is not variable unless the posted rules and/or pay table is revised.  Stated another way, all machines which play the same game and have the same pay table, must have the same payback.

(D)  It follows that players can tell which machines are the most liberal and can learn the strategy to optimize the payback.
(E)  The essence of Video Poker strategy is that every hand must be played (cards held) in the way that the hand has the maximum win-potential.

(F) The win-potential of a hand is indicated by a numerical value known as EXPECTED VALUE (EV). Players do not have to remember exactly how EV is derived  or even what the EV of any hand is, but they have to know the proper way to hold/discard so that the EV is highest.

(G)  Once the deck and paytable are defined, a ranking table is available in Video Poker books which shows the way to play every hand that can be dealt and played in that version.

            Learning the ranking tables is a lot easier than you might imagine since most hands are playable in only one way, which is obvious.

            We'll continue this treatise soon; in the meantime, practice on the kitchen table by dealing out 10 cards, five down and five up on top of them. That's how the machines do it. Rember that the caveat "Play With Your Head" translates into "Learn How First".