Tuesday, April 24, 2012

Jackpot Power

            As we get deep into the political season, we're all going to be frequently reminded how it is possible to make numbers say just about anything we want them to.  Quite frankly, it is not just the arena of politics this happens in.  It can be done with all types of math - casino math, included.

            By now, many of you well know that a full-pay jacks or better machine pays about 99.5%, which is a very solid number for a casino game.  Many of you may even be aware that the Royal Flush contributes 2% of this amount.  But what does this really mean?  It means that if the machine was defective and NEVER dealt a Royal Flush, but dealt all the rest of the hands in the frequencies we would expect, the payback of the game would be closer to 97.5%.   This is about the same payback we would get from a short-pay (8/5) jacks or better machine so should we expect roughly the same experience?

            ABSOLUTELY NOT!  One of the measures I like to use is what I call a 'session simulator'.  This process simulates a session of play for a particular game.  For video poker, I use 3 hours of play at 700 hands per hour.  For this particular demonstration, I ran 1000 of these sessions under 2 conditions.  The first was a full-pay jacks or better machined that NEVER paid a Royal Flush.  To be clear, the only way this could ever really happen would be if the machine was broken or rigged.  As I don't believe the latter happens in any reputable casino, nor would a broken machine likely stay on the floor for this many hands - this is merely for illustration purposes and to prove a point.

            In this scenario, the Player still managed to walk away a winner about 28% of the sessions.  This compares to about 29% when a regular full-pay jacks or better is played.   Why is there such little impact to this?  Under normal circumstances, the Royal would hit only about every 20 cycles or so.  Some of these cycles would already be winners, so the Royal Flush doesn't change this.  It only changes the magnitude of the win.  In the cases where the session was about to be a loser, the Royal most likely flipped ONLY these into winners.  However, when we look at the long run, the overall payback of ALL the sessions put together was where we expected it to be - at about 97.5%

            When we put the 8-5 jacks or better machine (with the Royal occurring as it should), we find that the Player wins only 14% of his sessions.  His winning sessions are cut by half!  The overall payback of all the sessions is also what we would expect it to be at 97.5%.

            So, why do two different machines paying about the same amount create such different short-term results?  This goes to a concept of volatility.  There is a mathematical formula for volatility, but I'm afraid if I start explaining it at that level, you're all going to turn the page.  That is why I like to use the session simulator as a means of explaining what volatility does and is.  When a large amount of the payback is concentrated into a very infrequently occurring hand, there is a larger degree of volatility.  In the case of the full-pay jacks or better game without the Royals, I removed a large degree of the volatility.  This is why a game with a considerably lower payback that the original version can still have a not very different short-term result.

            So, what does this all mean for you?  There are two points I'd like you take away from this week's column.  The first is to realize how important the Royal Flush is to your long-term results in video poker.  If you are on a cold streak of Royals, your short-term results may not look all that different from 'normal', but you may find that your larger bankroll is suffering.  If you play for 3 hours at a time, you may find that you're still leaving the casino a winner 3 out of 10 times, but for some reason your wallet still seems a lot lighter than it should.  The good news is that in the long run, those Royals will show up as often as they should (assuming you are playing Expert Strategy).  Ironically, when the Royals are running hot, you'll still walk away a winner about 3 out of 10 sessions.  But, a few more of those sessions will be big winners.

            The second point I want everyone to think about is if a 'mere' 800 unit payout occurring roughly every 40,000 hands can make this type of impact to a game, imagine what happens on a slot machine that can pay hundreds of thousands or millions of dollars for a 'hand' even more infrequent.  The average payback on a slot machine is ONLY 92-93%.  If we consider that many of them will have a massive top pay that might occur only every few hundred thousand hands (or million hands), what % of the overall payback does this account for? 

            With these occurrences being so infrequent (and COMPLETELY unknown as to how frequent), the payback of the machine without the jackpot could easily be 80-90%.  I'd put this through my session simulator but as it is not possible to know the frequency of all the payouts, there is no way to do it.  Just for fun, I built an 82.5% video poker paytable and put it through the process and it showed that the Player will walk away a winner only 5% of the time.  As we've already shown, it would then be possible to create an infrequent, very high paying jackpot which will push the overall payback up, while barely changing the short-term results. 

            The end result is one that we know all too well for slots.  Very few people walk away a winner even in the short run, which pays for the handful of people who win the big jackpots.  I'll take video poker any day!

Tuesday, April 17, 2012

Deuces Gone Wild

            I love getting fan mail and/or e-mails from readers.  There are two reasons for this.  The first is that it is always nice to know that someone is actually reading my column.  It is especially gratifying when someone tells me that they ALWAYS read my column.  The second reason is that a question from a reader can frequently become the basis for a particular week's article.  There are times I sit down to write my column and I simply don't know what I want to write.  I think of a topic and realized I covered it at some point.  Of course, being that I have now been writing for Gaming Today for more than 8 years, it is possible that I last wrote about the topic in 2005 and by now there may be some new readers.

            This week, I received an e-mail from someone who was questioning some of the strategy for Full Pay Deuces Wild.   Full Pay Deuces can be found in several casinos in Las Vegas.  As is the case for most 100+% machines, you won't find them on the strip.  You're going to have to head to some of the local casinos (such as Station Casinos) if you want to find them.  My source (www.vpfree2.com) shows that there should be 100+ machines scattered about at a variety of denominations up to a quarter. 

            While I've never been a big fan of Deuces Wild, this is just a personal choice.  Any game that pays 100.6+% is hard to criticize and is a good game for the regular Player to learn and master.  The strategy table is rather long, but when you break it down by the number of wild cards, you realize that it is not really a hard strategy to learn.  With a paytable that begins paying at Three of a Kind, you don't have to worry about counting High Cards.  The one thing, I strongly advise the beginner to learn is how to recognize hands with lots of wild cards in them.  It can become very easy to not realize that 2 6D 9D 10D KH is a 4-Card Inside Straight Flush with 1 Wild Card. 

            The question I received this week was specifically about holding a 4-Card Inside Straight (presumably with no Wild Cards) versus throwing all 5 cards as a Razgu.  The strategy table tells us that we hold the 4-Card Inside Straight.  If you look at the strategy table in Winning Strategies for Video Poker, however, it lists both hands as having an expected value 0.3+ - although it does list the 4-Card Inside Straight higher meaning its "+" is greater than the Razgu "+".

            While I write extensively on Video Poker in Gaming Today, I spend most of my time analyzing table games.  Many years ago I did write my own video poker engine that allows me to analyze most video poker paytables.  One of the limitations is that it does NOT do wild card games.  Fortunately, I have both other resources available to me and the ability to quickly create a program to help determine exactly how much those "+" are worth.

            Calculating the expected value of the 4-card Inside Straight was very easy.  There are 8 ways to draw the Straight (4 Wild Cards plus 4 of the 'natural' way to complete the Straight).  Each pays 2 units so we have a total return of 16 units.  Divide this by 47 ways to draw and we have an expected value of 0.3404.

            The Razgu is a bit more complicated.  As I've written about in the past, the overall expected value as shown in a strategy table for a hand like a Razgu is the actually the AVERAGE of all the possible hands of that type.  Often, no single hand will actually have EXACTLY the expected value shown. 
            About 20% of all hands in Deuces Wild are classified as a Razgu, each with their own subtleties.  The exact make up of suits and ranks will have some impact on the exact expected value.  For each 10 through Aces that is in the hand, there will be less chances to make a Natural Royal.   The exact suit composition of the initial deal will impact the number of possible Flushes that can be made if we discard all five cards. 

            In this particular case, however, the reader was talking about a 4-Card Inside Straight, which does limit the possibilities.  In order to get a more exact expected value, I quickly set up a program that had the initial deal set to 3D 4C 5H 7S 8S.  I figured that by leaving in all of the High Cards I would leave the expected value about as High as it could go and we could see just how close of a decision this really is.

            The expected value of this specific Razgu came back at 0.3267.   So, it would be more accurate to say that a Razgu is about 0.33- and a 4-Card Inside Straight is 0.34+.  It is not exactly a canyon between the two expected value, but there is a clearly superior choice.

            To help me prove my work, I realized that we also sell a Deuces Wild tipsheet that my father created a long time ago.  It has more detailed numbers on it.  It actually lists the expected value to two decimal places.  It lists the 4-Card Inside Straight 0.34 and the Razgu at 0.32.  (When all the possible Razgus are considered, the average must wind up at below 0.325).   It was good to know that my quick and dirty program was able to produce accurate results!

            If you're interested in learning the strategy on Deuces Wild, we have the tipsheet for $2.95.  It includes the strategy tables for Deuces Wild, Double Pay Deuces Wild and Triple Pay Deuces Wild.  Or you can order Winning Strategies for Video Poker which includes these 3 paytables plus dozens more for only $5.  Send a check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89128.

Tuesday, April 10, 2012

Sometimes, Is Less More?

            This week, I received an e-mail from a reader who was interested in a game called Triple Bonus Poker.   He enjoyed playing the one and only machine he could find of it in Las Vegas, but was unsure of the payback and strategy.  Not familiar with the game off the top of my head, I went to check my copy of Winning Strategies for Video Poker and found the game the reader was talking about.  He had actually found a Full-Pay version of the game and its payback was a very respectable 99.6%.

            What also quickly struck my eye was that the strategy table was much shorter than most others.  Then I noticed the top of the page which said "KINGS or BETTER".   Triple Bonus Poker doesn't pay on Jacks or Better, it only pays on Kings or Better.  Yes, the payback is still 99.6%.  It does this by paying well for Quads (240,125,75 - no kickers required) and VERY well for Full Houses and Flushes - 11 and 7 respectively.   It should be noted that it only pays 1 for a Two Pair, so this game is going to be VERY streaky.

            So, why play this wild game?  Well, that very short and easy to learn strategy table is what intrigues me.  I've often wondered what the error rate is for many Players given the intricacies of the standard jacks or better strategy table.  Keeping track of those High Cards makes for a long strategy table with subtle differences between 4-Card Straights and 3-Card Straight Flushes.   Looking at the strategy table for Triple Bonus and most of it is fairly intuitive.  Yes, it helps to know for sure that you throw a Full House if you have Three Aces and you dump Two Pair if you have a Pair of Aces, but these are easy things to remember. 

            The strange part is that despite having a respectable payback and a relatively easy strategy table, my source for games in Las Vegas tells me that NONE of these games exist.  As this source is user maintained, it is on occasion incorrect as it is in this guess.  There appears to be at least ONE table in Las Vegas.   Easier to find is a game called Triple Bonus Plus or Triple Play Plus. 

            One has to be very careful to not confuse these two games as their names have more in common than the actual games themselves do.   First of all, the latter game is a Jacks or Better game.  This means a full-length strategy table.  The Straight Flush is upped to 100 from 50, but the Quads pay of 75 is lowered to 50.  Most noticeable is that the payouts for Full House and Flush are the more pedestrian 9 and 5, respectively.   The end result is a payback of 99.8%.

            A game that pays 99.8% can actually be profitable for a Player when you include cashback and comps.  Or, at the very least, it can be a neutral game which you can play for very long periods of time with a relatively small bankroll.  I'm not one to dismiss this idea.  Also, while 99.8% and 99.6% might seem very similar, I am frequently the one to point out that you should at this from the other side.  One has a 0.2% house advantage and the other a 0.4%.  In other words, Triple Bonus Poker has TWICE the house advantage of Triple Bonus Plus.

            That all said, Triple Bonus Poker offers the Player a relatively easy game to learn without all of those pesky High Cards.  I have little doubt that for the average Player, the error rate will go down and the gap between the two games will be reduce to below the 0.2%.  If you are truly an expert Player, this will matter less to you.

            Of course, it would seem that the casinos have taken the choice away from the Players anyhow.  While there is perhaps a single machine of Triple Bonus Poker in all of Las Vegas, my source states that Triple Bonus Plus can be found in moderate abundance - at least in some casinos that target locals.  This part isn't a surprise because the best paying machines are generally found in the locals casinos.  What we don't know is whether the casinos removed the Triple Bonus games because Players didn't like the very streaky (and lower hit frequency) Triple Bonus or because the real paybacks were higher due to a lower error rate.  If this is what happened, it may have been a case where less was more for most Players.

Wednesday, April 4, 2012


            As I'm writing this, the country is beating itself into a frenzy not over politics but over a lottery.  The Mega Millions Lotto has an estimated prize of $640 MILLION.  That would make it the largest jackpot in the world.  Lotteries tend to have paybacks of about 50-60% so they aren't exactly a wise wager.  Yet, as I have often written, people are more willing to wager in games with bad paybacks if the top prize is life altering.  I think more than half a billion dollars meets that requirement.  I have to admit that if Nevada participated in Mega Millions, I would've tried to get some tickets.  I was NOT motivated enough to drive to nearby California to get them, however.

            Even when your choice of game is something like a Lotto, I think you should go in with your eyes open.  The odds of winning the top prize is about 176 million to 1.  To put that into a casino perspective, that is a little higher than the odds of being dealt a sequential Royal (10-A or A-10) in SPADES on the deal in video poker!  Of course, even if you're playing a Reversible Royals video poker machine, you're only going to get paid maybe $40,000 for that hit, not $640 million.

            Unfortunately, unlike most casino games, it is a bit more difficult to determine the expected value of this week's drawing for one major reason.  The $640 million dollars will be SPLIT by each of the people who have a winning ticket.  The lottery has stated that $1.5 billion worth of tickets have been sold, but from reading further it would appear that this is the TOTAL number of tickets sold since the last time the jackpot was won.  This does NOT represent the number of tickets sold for this particular drawing which is all that matters.  If we actually knew how many tickets were sold for this drawing, we could determine a more accurate expected value. 

            Armed with this information - and $176 million, it might actually pay to buy every possible combination of numbers.  We would then actually be wagering on how many other people hit the same set of numbers.  If less than 3 others, we would actually make some money on the deal.  Well, BEFORE Uncle Same takes hit cut anyhow.  To really make money, we'd probably have to be the only one to have the winning ticket.  History tells us this is unlikely and even less so if you were to add in someone who bought EVERY ticket. 

            No one plays these types of lotteries believing it is a wise investment.  We all know that the odds are very long.  The payback of the lottery is normally around 50-60% and even when it grows this large, it is probably no more than 70-80% when we consider that we are likely going to have to share it should we get struck by lightning and actually win.  What I find most amusing about these situations is the comments we get from some people.

            Today, I was reading a rather whimsical article about just how much money the $1.5 billion that was spent on lottery tickets really is.  It talked about how many families it could feed and how many trips to the space station you could make with this type of money.  Sadly, it also explained how it was only 0.1% of the national debt.  The article then moved on to quote some people who chose to play and why.  Of all the things I read, the one that made me to a double take came from an accountant in Louisiana (I won't post his name here):

            The article stated that the gentleman had bought 55 tickets and that he knows buying that many tickets doesn't mathematically increase his odds, and that his $55 could have gone elsewhere. He spent it anyway.

            "Mathematically, it doesn't make a difference, and intellectually we know that. But for some reason buying more tickets makes you feel more lucky," the accountant said. "Even people who know better are apt to feel that way."

            Maybe, he bought 55 tickets all with the same numbers?  Mathematically, buying more tickets doesn't make a difference?  So, if I buy one ticket I have the same chance to win as someone who buys 2 tickets?  What about the guy who buys 10?  or 50?  or 55?  or 176 million?  As an accountant, I would think he would understand numbers a bit better.   If you buy 2 tickets your probability of winning doubles as compared to buying 1 ticket.  If you buy 10 tickets your probability of winning multiplies ten-fold.  This gentleman bought 55 tickets, so he brought his odds down to a mere 3.2 million to 1 of hitting the big jackpot.  

            Fortunately, playing the Lotto takes as much skill as playing slot machines.  But many of the same people walk into a casino armed with about the same level of knowledge of the games.   Yes, mathematically, we know in the long run that we are likely to lose, but that doesn't mean we should take prudent steps to keep our losses to a minimum and give ourselves the best chance to win in the short run.  Because, in the end, mathematically, it all makes a difference.