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".

Tuesday, November 29, 2011

Giving Thanks

            I apologize to those of you who have been looking for my column the past couple of weeks and couldn't find it.  As some of you may have heard by now, my mother (and wife of Lenny Frome), passed away two weeks ago.  After the funeral, my brother recounted a story to me that I had never heard before.

            When my father passed away in 1998, my brother was the first one who headed out to Las Vegas to be with our mom.  It took a day or two before all the arrangements were made for them to come back East for my dad's funeral.  Yet, of course, they still had to eat.  My brother asked my mom where she wanted to go to dinner and she responded with Hugo's Cellar at the Four Queens.  My family had already made that a regular dinner spot when anyone came to town - and it is a tradition that carries through until today. 

            As they walked through the casino from the parking garage to the restaurant, they passed by two women playing video poker.  They were each holding a copy of one of my dad's books.  My brother said he could not have staged it any better if he tried.  This was clearly a sign.  My father's impact to the industry would continue long after he was gone.

            My father was informally called "the Godfather of Video Poker" by many in the industry.  To be sure, he played NO part in the invention of the game.  At the same time, no one can deny the impact he had on popularizing it.  Even if you are not a video poker expert or even a regular, I can't help but imagine that your play isn't just a tiny bit better from having read his articles - or any of the numerous writers who came after him - including me!  Would video poker have had the staying power if there wasn't someone telling the early Players how to play it?  Would video poker have eaten up as large a percentage of the casino floor as it does today?

            Of course, my father could just have easily been called "the Godfather of Proprietary Table Games".  He had a hand in the development of Let It Ride, Three Card Poker, Spanish 21 and Caribbean Stud Poker.  At their respective peaks, there must have been a combined 2500-3000 of these tables.  As I consider myself an extension of my father's work, we can add on Ultimate Texas Hold'em, Mississippi Stud and a host of smaller games to the total.  This brings the total to perhaps as high as 4000 proprietary tables that my father directly or indirectly had a hand in.  Imagine the casino floor without any of these games.

            While my father was the public face of everything that went on, everyone that knew them (both personally and professionally) knew that my parents were always together.  My dad brought my mom to business meetings to size up the potential client.  My mother was the proofreader for all of my dad's books and booklets.  She was responsible for shipping orders and for the accounting.  In fact, it was my mother who was always listed as the "President" of their company. 

            With the help of Catherine Jaeger, the editor of Midwest Gaming and Travel, we have launched a campaign to get my father into the American Gaming Association's (AGA) Gaming Hall of Fame in 2012.  No disrespect to Blue Man Group (one of the inductees for this past year), but I truly believe Lenny Frome's impact on the industry has been far greater.  To this end, we are asking people to write to the AGA and urge them to induct my father into the Hall of Fame in 2012.

            There are a number of ways to make your voice heard.  You can copy the sentence below or use your own experience to explain why you believe the time has come for Lenny Frome to be inducted into the Gaming Hall of Fame. "Because of his many significant contributions to casino gaming, I respectfully request your consideration of Lenny Frome for induction into the Gaming Hall of Fame."

Mail it to:
American Gaming Association
Frank J. Fahrenkopf, Jr., President/CEO
1299 Pennsylvania Avenue, NW, Suite 1175
Washington, DC 20004

E-mail to:
Brian Lehman/Communications Manager-AGA

            Over this Thanksgiving weekend, my family and I one again dined at Hugo's Cellar.  This time, for the first time we toasted the memory of both my father and my mother.  My dad may have been the "Godfather of Video Poker", but most importantly, they were the "Father/Mother and Grandfather/Grandmother of the Frome family."  Once again, they are "always together."

Tuesday, November 8, 2011

Beating the Casino

            A couple of weeks ago, I discussed how a blogger lamented how the last good table game invented was blackjack because none of the games invented since gave the Player a chance to beat it.  I would argue that blackjack is hardly 'beatable'.  It requires an incredible amount of discipline and knowledge of card-counting schemes in order to eke out even the slightest edge.  With the continuous shufflers and regrettably 6 to 5 payouts on blackjack, even with the most sophisticated counting scheme, beating the game is almost impossible.

            There is only one game on the casino floor (barring the occasional game that accidentally makes it to the floor) that is readily beatable.  That is video poker.  There are games all are over Las Vegas with paybacks of over 100%.  If you learn the proper strategy you can earn these paybacks.  I will not, however, promise that you will become filthy rich off them.  Unlike card-counting schemes which would allow you to bet $5 when the count is against you and perhaps $5000 or more when it is in your favor, video poker machines essentially have a constant wager and there are no known counts for you to track.  Each hand is completely random and no matter how many hands you have won or lost in a row, the next hand is still random.

            Over the years the casinos have caught on to this idea and this is why many are still willing to put machines on their floor with paybacks over 100%.  They simply don't put them on the floor with high denominations.   If you are willing to play video poker 40 hours per week (like a job), are able to play 700 hands per hour, you would play 1,456,000 hands per year (approximately).  If you played a quarter machine, you would wager the staggering sum of $1.82 Million per year.  If you can play the entire time on a full-pay Deuces Wild machine paying 100.76%, you'll earn about $13,800 per year.  This doesn't account for any taxes and doesn't account for any cash back and/or comps.  

            Of course, some of you will just suggest playing a higher denomination.  Even playing dollars, the annual 'salary' will get to only $55,000 or so.  This is certainly not a bad income.  Of course, this most certainly doesn't mean you can count on any sort of regular 'paycheck'.  There are going to be weeks you lose and there are going to be weeks you win far more than average.  And, as I said earlier - the casinos have caught on to this.  According to my research, there are no $1 machines playing full-pay Deuces Wild in the Las Vegas area.  So, the casinos are willing to let the resourceful Player make some money, but not enough to really entice large numbers of Players to do them harm. 

            This is also further proof that video poker machines are not slot machines.  Casinos would NEVER allow banks of 100+% slot machines to exist.  Slot machines require no strategy.  Thus, as long as the slot machine were to be played for 40 hours per week the casino would be paying out the $13,800.  To the casino it doesn't matter if this is one person, 10 people, 100 people or 1000 people who wind up winning this money.

            Video poker requires that the Player learn the proper strategy to earn this money.  So, if 52 people took turns playing for 1 week and played using Expert Strategy, then the casino would still be paying out the $13,800.  But the casinos know this is highly unlikely.  Despite strategies for virtually every imaginable game being available for 10-20 years, the overwhelming percent of Players simply choose to ignore the proper strategy.  The reality is that 50% of the Players probably know of no real strategy and just muddle along - playing a Deuces Wild game at no better than a 95% payback.  Perhaps another 25% play using some rudimentary strategy that they learned somewhere and play at 97%.  Another 15-20% have made real attempts to learn strategy but haven't really mastered it and can play at 99%.  The final 5-10% have learned the proper strategy and make a significant attempt to play properly.  Perhaps half of this group truly attains a payback of 100.5% or better.

            The result for the casino is that their machine probably pays out at no more than 96.5% in total.  A number that they can definitely live with.  However, just as the casino doesn't care if it is 1 or 1000 Players that win the $13,800, you do not have to be concerned with how any other Players do.  The casino is very happy with that machine that pays back 96.5% and will net them more than $65000 per year - EVEN if it means one of the Players made a few hundred or thousand over the course of the year.

            But, if you want to be this Player, you have to learn how to play video poker using the proper strategy.  To help you along the way, we're offering up our two full-length books for just $5 each (which includes postage and handling).  Winning Strategies for Video Poker includes strategy tables for 61 of the most common games found anywhere.  America's National Game of Chance: Video Poker is 200 pages of Lenny Frome's best articles, stories and quizzes and is an excellent way to learn how to play video poker in an easy to understand way.

            Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133 and you'll be on your way to becoming an Expert Video Poker Player.

Thursday, November 3, 2011

The Godfather of Video Poker

The November issue of Midwest Gaming and Travel is a tribute to my father, Lenny Frome.

I'd like to thank Catherine Jaeger (the editor) for coming up with this idea and using it as a means of launching a campaign to have my father inducted into the American Gaming Association's Gaming Hall of Fame.  Here is a little bit more about this campaign - Make 2012 Lenny's year!

My father changed the landscape of every casino in the world by helping to make video poker as popular as it is.  He also provided the original analysis for games like Let It Ride, Three Card Poker, Caribbean Stud Poker and Spanish 21.  Imagine the casino floor without any of these games.

Below is a link to the article I wrote for Midwest Gaming and Travel about my dad.

The Godfather of Video Poker

Tuesday, November 1, 2011

Video Poker Progressives

            A couple of weeks ago, I described in detail how the math behind Progressives work.  In that column, I mentioned how video poker progressives works just a little different.  There are still two paybacks to be concerned with - the long term theoretical that the casino is concerned with and the specific payback at any point in time that should be the attention of the Player.  The majority of the calculation is still the same in that we multiply the payout of a winning hands by the frequency of the winning hands  and sum up these values.

            What is different about video poker is that the frequencies of the different winning hands can vary as the amount on the meter changes.  For those of you who are video poker Players, this should be no surprise.  For years, I've been telling you that a single unit change in the payout of a hand not only changes the payback but can change the strategy.  Each time you change the strategy you potentially increase the frequency of some hands at the expense of others.

            As a very simple example of this, imagine how the strategy changes as we go from a standard full-pay jacks or better machine to a Double Double Bonus machine.   Because the payout for Four Aces is so high, we actually find that the Player should discard Two Pair in favor of a single Pair of Aces.  This will obviously reduce greatly the frequency of Two Pairs and Full Houses and increase the frequency of Three of a Kinds and Four of a Kinds. 

            So, it should be no surprise that as the jackpot for a Royal increases above 800 that the strategy will begin to shift.  Hands with the potential to be a Royal will have their expected values increase.  This will lead to more Straights, Flushes and of course Royals and the expense of Pairs, Trips and Quads.  Of course, we will also throw away a variety of partial Straights or Flushes to go for the Royal, so this will work against the Straights and Flushes and might increase the number of High Pairs. 

            Thus, pinpointing the exact frequencies can be a bit tricky.  Fortunately, the far easier of the paybacks to determine is the payback at any point in time.  This is because at any point in time, we can know the exact amount of the Progressive jackpot and use this number to determine the exact strategy and in turn the exact frequency of each hand.

            The most common video poker Progressive is an 8-5 machine, meaning it pays about 97.3% when the jackpot is reset to 800 (per unit wagered).  At this level, the frequency of the Royal is about 1 in 40,200 hands.  If the jackpot were to climb to 1600 (per unit wagered) then the payback of the game will go up to about 99.5% and the frequency of the Royal goes up to 1 in 32,700 hands.

            Of course it is rather unlikely that you're going to see a Progressive for a Royal get this high.  With only 1% of the amount wagered (at most) going to the meter, the average amount that will be added to the Progressive Jackpot is somewhere between $327 and $402 (1% of the previously mentioned frequencies).  Of course, something that can occur 1 in 40,000 hands or so can easily occur every 10,000 hand or 80,000 hands.  So, it is not impossible to see the progressive meter go to 1600.  It would have to go to about 1800 for the game to become positive (payback over 100%).  This is not impossible, but not very likely.

            As the payback goes up, the strategy changes and the frequency of the Royal increases, making it harder and harder for the jackpot to keep increasing as the likelihood that it gets hit goes up.  Because of this, it is a bit harder to calculate easily the long term theoretical payback.  It is reasonable, however to approximate it using the same process used for regular progressives.

            In this case, I would take the frequency of each hand using the reset value of the jackpot and multiply each by the payout of the hand and sum these up.  Lastly we would add the percent of each wager going to the jackpot to the total.  This means that the long term theoretical payback of a Royal paying 8-5 with an 800 unit reset amount is about 98.3%.

            I have to admit, if I were designing a paytable for a video poker progressive, I would probably make the likelihood of the game going over 100% a bit more common.  I think it would be a lot of fun to the frenzy that would/should occur each time the payback at any point in time goes over 100%.


Friday, October 28, 2011

Vintage Lenny Frome - A's and 8's

When I have time I'm going to try and post up some of my father's (Lenny Frome) articles here as well.  The following article is about a rarely found (but I'm told there are still a few 50-cent machines at Circus Circus) version of video poker - Aces and Eights.  Its payback is about 100.25%.  This article is probably 15-20 years old, so some of the information may be dated:

Aces and 8's--From Green Felt to Video  

             Long, long ago before the world played Video Poker, the story of Aces and Eights, the dead man's hand was abroad in the land. It always conjured up a mental image of evil-- a hand that brought fear into the hearts of men, even the roughest, who made a living with the pasteboards and reckless gun slinging.

             Leave it to the enterprising folks at CircusCircus to capitalize on this theme and then carry it out in high-tech fashion under their big top. They have come up with a sure winner in Aces and Eights, a 100% payback machine featuring four progressive Jackpots as added attractions which will frequently push the payback into positive territory.

            The pay schedule  is very straight-forward for a multi-progressive (or is it just that we are getting adjusted to the new regimen in such lengthy tables?)  It is an 8/5 schedule Jacks or Better with these four bonuses to offset the 2.3% shortfall vis-a-vis full-pay 9/6ers:
·         Four 7's pay 50 for 1 on 1 to 5 coin-play non-progressive.
·         Four 8's or Four Aces pay on a single progressive which resets at 80 for 1 (5-coin play only).                
·         A Royal Flush pays on an 800 for 1 minimum progressive jackpot.
·         Sequential Royals (either way) pay on a 10,000 for 1  minimum progressive

Looking at the payback situation, these bonuses work into the picture this way:

            Four of a Kinds in any one specified suit occur on average only  once in 5,500 hands; a regular Royal once in 40,000 and in either    sequence once in 2,400,000 hands.  The extra 25 on the 7's adds .47%. The extra 55 on the 8's adds a  minimum of 1% as does the extra 55 on Aces. The Sequential Royal gives us an extra 9,200 which is worth .38%. As the meters climb  upward the value of these jackpots further increases the payback. Together, these bonuses,  take the payback up from 97.3% to 100.15%. With some minor changes in strategy, we can pick up a little bit more.

            The first time we saw this machine, the Sequential was posting $13,204, the Royal  $1,030 and the Four Aces or Eights a whopping $154. The game was close to 102% payback. While watching it, the jackpot was hit on 8's by one of the players seated at about 50 machines on the floor.  The Aces/Eights progressive had gone an unusually long time since the last hit. We cannot expect many such generous jackpots. Frankly, we were genuinely surprised when the meter reset at $100, since that is more than three times the 25 for 1 normally paid on quads.

            Even with this liberal machine we need all the savvy we can muster to play the game expertly. The extra value of Aces and 8's dictates these modifications in the ranking order of 8-pairs: Even at minimum meter value of $100 on a quarter game (80 for 1), the pair of 8's is better than a 4-low-card flush and at $120 beats those 4-card flushes with two high cards.

            Incidentally, the player who hit the $154 would have been right in breaking up 8's full to go for the quads. I'm happy to report that it wasn't necessary to wrestle with that problem--but if it were you, what would you have done?


Tuesday, October 25, 2011

Don't Be Foolish!

            It was almost a year ago that I launched this blog (  I was very nervous about launching it.  If there is one thing I've learned about the internet over the years is that pretty much any idiot can have a blog - and quite frankly, I didn't want to be 'any idiot.'  I'd like to think that the name "Frome" is the gold standard in the industry where math analysis is concerned.  To our credit, we have Three Card Poker, Let It Ride, Caribbean Stud Poker, Spanish 21, Ultimate Texas Hold'em, Mississippi Stud, Imperial Pai Gow and countless sidebets.  That's a lot of the casino floor whose math was done by Leonard Frome or Elliot Frome. 

            So, I was quite surprised this past week when I came across a financial blog that was very unimpressed this year's G2E where table games were concerned.  Admittedly, I did write a column last year that called on more inventors to display their ideas at the G2E.  I recognize that the cost of even a small booth can be rather prohibitive for the individual inventor, but what a great opportunity to show your game to people in the industry.  I was pleasantly surprised to see at least two new inventors displaying their games and larger booths from some of the more established companies.

            What I found amazing about this financial blog, however, was not that the writer looked over every game and found none of them to his liking.  That would've been one thing.  Instead he essentially takes table game companies to task for "designing games that the gambler has no hope of beating, but they force the gambler to take the time to learn how to play them!"  This blew me away!  Does he truly expect the casino to introduce games that the Player can easily beat?  That's not going to happen.  The only game that has ever been put on the floor that can readily be beaten are certain variations of video poker. 

            Further, our blogger is annoyed that you have to take time to learn how to play them.  The only game which requires ZERO time to learn how to play them is perhaps slot machines.  As I've recounted in my column many times, I can't even figure out when I've won or lost anymore in today's video slots, but since all you need to do is press the 'spin' button and we can assume that the machine will properly tally your win or loss, I assume this meets the requirement of not needing to take time to learn how to play them.

            Thus, we can conclude from our blogger that what he is looking for is a slot machine with a 100%+ payback.  Perhaps he should've read my column from two weeks ago where I talked about a company that provides the payback information for their slot machines.  This WOULD necessitate learning how to use the smartphone 'app', so I don't know if this meets his strict criteria.

            A couple of days after this first column appeared, our blogger was back with more information for us.  First, he repeats some of his thoughts from the previous column, decrying the lack of innovation from table game companies and then stating, "how the gaming industry has not seen a blockbuster table game since blackjack, and how the industry may not see one until somebody steps up and creates a game that is theoretically beatable."

            That is quite a statement.  According to Wikipedia, blackjack's origins may be as much as 400 years old.  The game as it is played in most jurisdictions is hardly beatable - or at least not easily.  Yes, we're all aware of the MIT team that did it, but this took a rather significant effort on the part of a focused group of individuals. 

            In 1991, the table half of the casino floor consisted of nothing but blackjack, craps and roulette.  Twenty years later, it is estimated that as much as 15-20% of the tables in the U.S. market may be those that were invented AFTER blackjack.  Twenty years from now, I have little doubt that blackjack will make up an even smaller percent of that floor.   Let's not forget that a blackjack table is essentially FREE to the casino while they have to pay to put a proprietary table game on their floor.

            As a gaming analyst - and one that focuses mostly on table games - I am keenly aware of the math of the games.  Most of the newer games that are being introduced have paybacks in the higher 98% to low 99% range.  Yes, they do require that you 'learn' how to play them to achieve these paybacks.  No one, not myself, not the inventors nor the casinos will try and let you believe that the games are beatable in the long run.  That does NOT, however mean that you cannot have winning sessions in the short run and enjoy the entertainment value that they can provide.  Most table games are developed to have the Player win about 35-45% of the time over a 3-hour session - assuming you are willing to 'take the time to learn how to play them'

            Best of all, I won't "force" you to do this, but I'll give you the opportunity to!  There are now 7 books in the Expert Strategy series for table games (Let It Ride, Three Card Poker, Four Card Poker, Spanish 21, Caribbean Stud Poker, Mississippi Stud and Blackjack Switch) and for a limited time, you can order the entire set for $20 which includes postage and handling.  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.

Tuesday, October 18, 2011


            Last week, I alluded to the seemingly complex math associated with games that offer progressive payouts (i.e. "progressives").  Progressives are games where the top pays are not fixed dollar amounts or odds payouts, but rather have variable payouts that increase as more wagers are made since the last time the prize was won.

            Progressives have become very popular for table games sidebets.  They have long been used for some video poker machines for payouts on Royal Flushes.  Most commonly they are found on slot machines, which love to use a progressives ability to create a very large payout for a very rare occurrence.  As is always the case with a random event, the cycle between hits can frequently become far larger than 'average' and thus create an even larger than normal jackpot.

            As I described last week, Progressives essentially have two different paybacks.  The first is the long-term payback which is what concerns the casino.  The second is the payback of the wager at any point in time which is what should concern the Player.  Let's take a closer look at how these are calculated and why there are two different paybacks.

            Normally, to calculate payback, we take the frequency of a winning hand, multiply it by the payout of this hand which gives us the contribution rate for the hand.  We then sum up these contribution rates to arrive at the overall payback.  For most wagers, the frequency of a particular winning hand is fixed as it is unaffected by strategy.  So, if we are playing Caribbean Stud Poker, we don't have to worry about the strategy of Folding and Playing for the sidebet because you would never Fold a hand that is strong enough to earn a bonus.  Video Poker presents an additional challenge in that you can alter you strategy depending on the payouts and thus alter the frequency of winning hands.

            So, to calculate the payback of a Progressive at a particular point in time, we follow the calculation I just described.  For example, let's assume the following paytable at a particular point in time for a $1 wager:

Pays (For 1)
Royal Flush
Straight Flush
Four of a Kind
Full House
Three of a Kind
Two Pair


            If we perform the calculation described, we get the following:

Pays (For 1)
Contribution Rate
Royal Flush
Straight Flush
Four of a Kind
Full House
Three of a Kind
Two Pair


            So, if you were to walk up to a table and see these payouts, the payback of the game at that very point in time is 81.46%

            But, the payback to the casino could be vastly different.  Let's assume that the Royal Flush is seeded at $50,000.  This means that every time someone wins the jackpot, the prize for the Royal Flush will be reset to $50,000.  Further, let's assume that for every $1 wager that is made, the Progressive increases by 10 cents (i.e. 10% of the wager).

            There are two changes that we must now make to calculate the payback for the casino.  The first is that we always use the seed amount as the payout for that hand.  Thus, we repeat the calculation shown above but we use $50,000 as the payout for the Royal Flush.  This is the amount that the casino itself directly paying out each time the jackpot is won.  When we do this, we find that the payback of this wager is 79.08%.

            However, we must now ADD to this payback the amount of each wager this added to the Progressive meter - in this case 10%.  Eventually this 10% will go back to a Player.  It might happen while the jackpot is at $50,000.10 or it might happen when it is at $120,000 or more.  From the casino's standpoint, it doesn't matter.  That 10% belongs to the players.  Essentially all the Players that don't win the jackpot are handing those dimes to the person who finally does.  So, when we add that 10% to the 79.08% we find that to the casino this wager really has an 89.09% theoretical payback.  Over time, the casino will keep 10.91% of every dollar wagered.

            So, if you were to play this wager while the Jackpot is $65,473, you would actually be playing it on the 'low side' of the average jackpot.  How big is the average jackpot?  To calculate that, we take the average number of hands between jackpots (in this case 649,740) and multiply it by that 10%.  On average the jackpot will grow by $64,974 before it is hit.  We add this to the seed amount and find that the average jackpot will be $114, 974.  At that point, the payback of the wager is the same as theoretical payback of the wager. 

            If the jackpot grows to be above $185,930 (which is very likely at times), then the payback of the wager at that point will actually be OVER 100%.  The only problem with this is that it will only be over 100% for the ONE person who actually wins the jackpot.  Everyone else will just be feeding dimes to the one person who wins.