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.

Tuesday, October 11, 2011

A Game Changer?

            For more than 20 years, my father and I have implored our readers to kick the slot habit.  The biggest reason for this is twofold.  First, the paybacks on slots are just too low.  If I recall correctly, the State of Nevada reported that the average payback on a slot machine last year was in the 92% range.  You just don't stand much of a chance.  The second reason is that you have no clue what the payback is of any given machine.  So, even though 92% is an average and there might be some games paying even as high at 98% (not very likely), you have absolutely NO WAY of knowing which machine is doing this.  For the moment, I'll leave off that playing a game with absolutely no strategy is not one that I find very appealing to spend my time at.

            Well, this past week at the Global Gaming Expo, I had a conversation with the founder of a company who appears to be trying to change some of these problems with slots.  I find his attempts extremely noble.  I'm not necessarily convinced that the casinos are going to support what he is suggesting - but time will tell.

            The company trying to revolutionize the slot industry is called "Pharos Gaming" out of New York City.  The gentleman I spoke to was Frank Abramopoulos and he is a Director there.  The concept is rather simple.  Every one of their "Open Label" machines is given a unique digital code that is on the screen.  A Player can then access a 'app' on his smartphone and enter this code.  This app will return to the Player significant information about the slot machines.  Most importantly, it will tell the Player the payback of the slot machine.  As the company's slot machines are all single-machine Progressives, it will use the current size of the jackpot to calculate the precise theoretical payback of the machine. 

            With this single concept, Pharos Gaming has eliminated at least one of the basic problems with slot machines.  Any player can now know the payback of a machine instantaneously.   This is certainly progress.  But, even if you know the payback of a game, I don't recommend that you start playing games at 92%.  However, because Pharos Gaming's Open Label machines are stand alone progressives, the true payback of the machine to a single Player is fully dependent on the size of that jackpot at any point in time. 

            So, in theory, a Player can walk up to a bank of machines, punch in each of the special codes and then only play the game that shows a current payback of over 100%.  This has been going on in the video poker world for years.  The only difference is that for video poker, you don't really need a smart phone, you just need to remember the payback of a few base paytables and learn how to approximate the current payback based on a simple formula (I'll cover this again in a future column).  But, progress is progress.

            Explaining the math behind Progressives has always been a tricky proposition.  I have worked on many sidebets for table games that utilize Progressives and I've been amazed at how few in the industry fully understand how it works.  I won't go through the details here, but the short version is that you essentially have two paybacks.  One is the long-term payback which is the one the casino is worried about.  The other is the specific payback at any point in time that the Player should be focused on.  In the end, all the people who play on the Progressive but don't win it are playing below the long-term payback and the person who finally wins the Progressive is playing above it - IF the jackpot grows to be larger than average (which will happen roughly half the time).

            In the case of Open Label, the Players that are oblivious to the feature that allows you to check the payback are essentially feeding the machine so that those that are 'in the know' can come along when the game goes positive and win the jackpot.  This works so long as you have the two classes of Players - oblivious and in the know.  If EVERY Player who ever plays these games became an 'in the know' player, the model would fall apart.  NO ONE would ever play the slot when the jackpot is near the reset value, as the machine would probably be playing at 92% or less.

            So, how does this entire system differ from video poker and its progressives?  From what Frank told me, the Progressives on their open label add a large amount of each wager to the jackpot (he didn't specify specifically how much).  Mathematically, this can increase the likelihood that the game will go positive - assuming the hit frequency of the jackpot is low enough.  The bad news is that the only way to accomplish this is for the base payback at reset to be fairly low.  This requires that they continue to get those oblivious players even while they are marketing towards the more knowledgeable ones.

            In the case of video poker, the payback at reset is considerably higher than its slot machine counterpart and the amount of money added for each wager to the jackpot is relatively small.  Thus, video poker machines do not often get much above 100% (if at all) unless the base game is very close to 100%.  At the same time, this means that the Player who doesn't hit the jackpot is not giving up quite as much to the one Player who does.

            I tip my cap to what Pharos Gaming is trying to accomplish.  Any means by which Players can gain more information about the game they want to play is a good thing.  Also, there is some merit (and profit) in chasing games with paybacks over 100%.  However, a slot is still a slot and your chances of hitting that jackpot will be based 100% on luck.  With video poker, an Expert Player can gain advantage over other Players by modifying his strategy as the jackpot goes up and increase his chances of winning it.

            Despite this change, I still implore you all to kick the slot habit and try video poker instead.  To help you along this path and to celebrate our arrival in Las Vegas, we're offering up some specials!  You can order any or all of our top video poker titles for just $6.95 each - Expert Video Poker for Las Vegas, Video Poker: America's National Game of Chance or Winning Strategies for Video Poker.  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133.  All prices include shipping and handling.

Tuesday, October 4, 2011

The Next Big Game

            Part 2 of my preview of table games at the 2011 Global Gaming Expo (G2E) will focus on two games that I did the math on.  Both of the games come from Shuffle Master and both with also be on display at their booth this year.   From what I understand, the 'live' table half of their booth will contain ONLY brand new games.  Up this week, Ultimate Three Card Poker and San Lo Poker.

Ultimate Three Card Poker

            If you only take a quick look, you may think this is just Three Card Poker.  If you look closely, you'll see a variety of twists have been added in.  The betting structure if vastly different.  To begin play, the Player must make an Ante AND Blind wager (equal aize).  The Player and the Dealer will each get their three cards, but now one of the Dealer's cards is turned face up.  This is a lot of information given it is just a 3-card hand.  As in the original Three Card Poker, the Player may now either Fold (surrendering BOTH his initial wagers) or make a Play bet.  If he has a Pair or Better, he may make a Play bet of 3x his Ante. 

            Qualifying still exists in Ultimate Three Card Poker, but the rules are a bit different.  If the Dealer does not have at least a Queen High or Better, the Ante pushes and all other wagers stay in action.  If the Dealer has a Queen High or Better, then all wagers are in action.  If the Player beats the Dealer, his Ante and Play win even money (unless the Dealer did not qualify in which case the Ante pushes).  The Blind will push if the Player beats the Dealer with less than a Flush and will win odds if he wins with a Flush or better.

            The critical thing to know about Ultimate Three Card Strategy is FORGET Q-6-4. Knowing one of the Dealer's cards make a whole lot of difference in what we Play vs. Fold.   I'll cover the detailed strategy at a later date, but I can tell you that Ultimate Three Card Poker affords the Player a strong 98.81%.  It will, however, require that you learn a strategy more complex than Q-6-4, but it is one that you can master easily.

San Lo Poker

            The first time I saw Pai Gow Poker, I thought it was some sort of high roller game, requiring a large bankroll.  When I found out more about it, it turns out it is likely the slowest game in the casino - at least from a bankroll perspective.  Around 40% of the hands end in a push with the Player winning 1 hand and the Dealer winning one hand.

            If you want to eliminate this condition, split the cards into 3 separate hands instead of 2 and require the Player to win 2 out of 3 hands to win.  Now, there is no such thing as a push.  To keep things more exciting, deal only 6 cards instead of 7 and you have San Lo Poker.  You can't get much easier than this.  The Player makes a single wager and is dealt 6 cards.  He must break it down into a 3-Card Hand, a 2-Card Hand and a 1-Card Hand (also called High, Mid and Low hands).  As in Pai Gow, the High Hand must outrank the Mid Hand which must outrank the Low Hand.

            The Dealer will also be dealt 6 cards and break his hand down according to the House Rules (which I helped develop).   As these rules are fairly complex, San Lo is played on tables with Shuffle Master's i-verify which has card recognition software built-in and will tell the Dealer how to set the cards.  A Player may opt to have his hand set the same way and can ask the Dealer for assistance to do this.

            The human playable strategy is still being developed, but at this point, I'm fairly certain that to maximize the payback, the Player does NOT always want to mimic the House Rules.  With a goal of winning two out of three, sometimes it pays to sacrifice one to help win the other two. 

            The payback of San Lo is just under 99% but there is no messy commission to deal with.  I'll cover this game in more detail in the coming weeks.  In the meantime, I suggest you stop by the Shuffle Master booth and check this one out.  It's an exciting twist that adds a bit more thinking to the classic Pai Gow Poker game.

            If you make it to the G2E this year, please feel free to drop me a line ( if you see any other table games along with any comments you may have.  I'll be wandering around the show floor for most of the three days, checking out the next big game!