Thursday, August 29, 2013

Think Loss Rate

            How much of a difference is there in terms of payback from one casino game to another?  Most table games have a payback between 97 and 99.5%.  Video Poker can range from about 95% to 101%.  Slot machines probably range from about 85% up to 95%.  Sidebets, quite frankly are all over the place, ranging from just over the legal limit of 75% and going up to the low-mid 90%.  While there is a lot of overlap, one of the largest determining factors is strategy.  More complex strategy means a combination of more human error and/or Players not even trying to follow it.  Simple strategy is much easier to learn and follow.  Three Card Poker has one simple strategy rule.  Follow it and you should approach the theoretical payback of about 98%.  Don't follow it and you can only do worse.

            Video Poker has paybacks considerably higher.  Not all of the versions, but you can still find plenty of them well above 98%.  Video Poker's strategy, however, is far more complex than Three Card Poker's strategy.  The average Video Poker machine has more than 30 different strategy items that need to be memorized and in the appropriate order so that you know how to play the hand.  So, first you need to review the hand and determine the realistic ways the hand can be played and then you have to know which of these ways has the highest expected value, which tells us which way the hand should be played.

            In most games, many of the hands are pretty obvious even if you knew little.  If you're dealt a 6-7-8 in Three Card Poker, I don't think you need to have read a book to know what to do.  What if you are dealt K-3-2?  What about Q-8-2?  What about Q-3-2?   For each hand, the Player is really asking himself if he is better off Playing or Folding.  Those are the only two options in Three Card Poker.  The answer is pretty obvious for the Straight and a good deal less obvious for the other three hands.  The strategy is determined by the math behind the question of whether the Player is better off Folding or Playing.  By Folding, the Player forfeits his original wager (one unit).  By Playing, he wagers an additional unit.  If Playing can return at least that additional unit (on average), then the hand is worth Playing.   The Player does not have to perform some complex calculation on each hand.  The decision is to Play or Fold and the math works out very neatly.  For every hand stronger or equal to Q-6-4 the Player is better off Playing.  For Q-6-3 or less, he is better of Folding.  You've just become an expert at Three Card Poker strategy.

            Video Poker is not nearly this simple.  First of all, there is no folding and no additional wagers.  You make an original wager and your only goal is to maximize the amount of money you get back on average for each hand.  If you're dealt a Straight off the deal, there isn't much to think about - unless of course it is also a 4-Card Straight Flush or a 4-Card Royal - then what?  What if you're dealt Three of a Kind and 3-Card Royal?  How about a Pair and a 4-Card Flush?  Does it matter if it is a High Pair or a Low Pair?  (Yes, it does!)  

            In Video Poker, the hands are categorized into about 30-40 different hand ranks and partial hand ranks.   Each of these is assigned an expected value.  This expected value is calculated by looking at ALL the possible draws for that hand and tabulating the total units won for each final winning hand.  We then divide this total by the number of possible draws so that we can compare apples to apples.  So, to look at a simple example.  Suppose you are dealt the following hand:

4♥        5♥        6♥        7♥        8♦

            The decision here should NOT be driven by your favorite Clint Eastwood line ("are you feeling lucky, punk?").  It should be driven by the math.   The straight has an expected value ("EV") of 4.00.  There is no draw in this case and the EV is simply the payout of the hand.  If you decide to discard the 8, there are 47 possible draws.  2 will result in a Straight Flush, 5 will result in a Straight (remember that you would have discarded a card that could also have made it a Straight) and 7 that will result in a Flush.  All other cards result in a losing hand.  So, do you throw away the sure 4 units to go for the Straight Flush?  When we add up the payouts of the winning hands, we get 162 units (2 x 50, 5 x 4, 7 x 6).  We divide this by 47 (the number of possible draws) and get 3.45.  As this is less than the EV of the Straight, we keep the Straight.  In the long run, this will be the better move.

            While most Player would play this correctly (I guess?), the simple reality is that except for those that learn the right strategy, there will be a significant number of Players who will NOT play this correctly.  Throw in the roughly 25% of hands that require a real decision and the casinos can count on Player error to help pad their winnings.  This is why they can offer the 99.5% paybacks on so many full-pay jacks or better Video Poker.   Someone like myself might sit down and get the 99.5%, but the vast majority of Players will play well below this level.   They are likely to play in the 97-98% range if they have some idea of what is going on and perhaps as little as 95% if they just 'wing it'.   The difference between 99.5% and 96% may not seem like a lot, but I always suggest you turn that around to the loss rate - 0.5% vs 4%.  Now there is a 700% increase from one to the other.  The impact to your bankroll could be staggering.

Thursday, August 22, 2013


            I consider myself to be a very competitive person.  Anybody who has ever played against me in a board game or on a sports field is pretty aware of this.   I play fair and hard.  I'll never cheat and don't throw tantrums.  But I really hate to lose.  So, you can only imagine what I feel like when I'm having 'one of those nights' while playing video poker.   Gambling isn't exactly the type of thing one does if they hate to lose.  Even if you're playing video poker or blackjack, games that are near 100%, you're still going to lose more than 50% of the time over short sessions.  Not a bad record if you're the Marlins, but I prefer to win, well, closer to 100% of the time.

            When I'm on the sports field, I have a significant amount of control in the outcome.  If I'm playing tennis, well, it is just about all on me.  If I'm playing softball, I can do my best to get on base when I'm at bat and make all the plays that come to me.  I can't help my right fielder catch the ball, however.  In this regard, gambling is more of a team sport than a single Player sport.  I'm an expert at just about any game in the casino that I will sit down to play.  So, I can make sure that I'll play each hand the way I should to maximize my overall payback.

            Unfortunately, luck still plays a significant portion of casino gambling (kind of like my right fielder catching the flyball?).  I can't control which hands I'm dealt.  In the long run, I know I will get my fair share of each type of hand.  In a given night, the difference between winning and losing is about getting your fair share of key hands.  You're not going to make money off of 4-Card Straights, so you don't usually keep track of how many you got. 

            When we look at the final paying hands of video poker, it should be no surprise that most of the payback comes from the bottom 3 hands.  Jacks or better gives us about 21-22% of our payback.  Two Pair gives us 26%, and Three of a Kind gives us another 20-21%.  This is almost 70% of a total of 99.5% payback.  Straights give us over 4%, Flushes over 6% and and Full Houses around 10%.  That brings us to 90%.   Four of a Kinds give us about 6%, Straight Flushes a mere 0.5% and Royal Flushes the remaining 2%. 

            The more common a hand is, the more likely no matter how weird your session is going that at the end of it, you're going to have very close to the number of those hand that you are supposed to.   So, if you play 3000 hands and the average shows that you should have about 650 High Pairs, you're not going to find out that you only had 500 of them.  Maybe you have 630 on a bad night and 670 on a good night, but you'll get very close to the 21-22% payback you are supposed to.

            On the other end of the spectrum is the Royal Flush.  If you play 3000 hands, you're well below the roughly 40,000 hands it takes to play to catch a Royal.  If you play a session and miss the Royal, you're inherently playing at 97.5%.  If you hit one then, well, you're assuredly playing well over 100%.   As a result, there really isn't a lot to discuss where the Royal is concerned.  It is literally hit or miss.  Straight Flushes simply don't add enough to the mix and are also so rare that you can't really look to them for a good or bad night.

            The critical hand is the Four of a Kinds.  Earlier I said that they make up 6% of the payback.  That is on a jacks or better game.  Move to Bonus or Double Bonus or Double Double Bonus and these number goes way up.   You win or lose in these games based on two key factors.  Do you get your fair share of Quads and which Quads do you get (when playing the bonus games)?   If you play 3000 hands, you can 'expect' to hit about 7 Four of a Kinds.  It would not be uncommon to play this many hands and get only 2 or 3.  If you have one of these nights, you're not likely to walk out a winner.  Quite frankly, you may not walk out with any of your bankroll left.  Fortunately, it is just as common to get 10 or 11 of them.  In these cases, you are very likely to walk out a winner.  If you're playing Double Double, you'll also want to hit some of the bonus Quads and/or the 'double' bonus quad with one of the kickers. 

            Playing the right strategy is, of course, a critical component of getting your fair share of Four of a Kinds.  But, the right strategy does only so much to make the 5th card in Quad 3's also be a 2, 4 or Ace.  Sometimes it just takes luck to have that good night.  Sometimes my right fielder actually catches the ball.  All I can do is hope.

Friday, August 16, 2013

Nutritional Labels for Casino Games?

            Today's column topic comes from my Freshman college roomate.  He posted a question to one of my old columns on my blog (  He wanted to know if I thought if the casinos would ever have to disclose all the key statistics about each game - a sort of 'nutritional label' for each game.  My response was that I didn't think so for 3 reasons.  The latter 2 were more political than mathematical.  This column is about that first reason.  With the exception of slot machines, all that information about each game is already fully known.

            While admittedly, if the average person were to walk up to a game that he has never seen before, he isn't going to know what the payback or win frequency is.   I'm an expert and I couldn't necessarily tell you these key stats about a game I've never seen before.  I might be able to take a good guess about it, depending on whether we are talking about a complete game or a sidebet.  There isn't a lot of variation in table game paybacks.  There are probably very few that are below 97% and, of course, none above 100%.  Sidebets can have a much larger range, as some are as low as 75% and others go up to the mid 90's or even a smidge higher.   If I were to walk up to a video poker machine that has a foreign paytable, but is based on one of the better known games, I can probably peg the payback to within .25% by doing some quick math in my head.

            There is little doubt that putting the key statistics on each machine would make this information far more readily available than the way it is currently done.  But, I wouldn't equate this to a can of soup without a nutritional label.   The list of possible ingredients and the exact quantities in the soup is nearly endless.  Throw in the fact that there are hundreds of thousands of food products (or more?) and it is completely impossible to make a choice based on nutritional content without these labels.  When you walk into a particular casino, you have perhaps a dozen or so choices of which table game to play.  Yes, each casino may have its variation of rules.  One may offer a 6-deck shoe for blackjack and the other may have a single-deck game.  One may hit on soft 17 and the other may stop on all 17s.  But, if you spend time reading a book or two on gambling, you'll quickly learn and likely remember the paybacks of most of these rule variations.

            For many games, there is almost no variations available - especially for the base game.  Want to play Three Card Poker?  It has a 97.98% payback for Ante/Play.  While there are some variations of Pair Plus, the overwhelming number of them have the same paytable, paying 92.72%.   These numbers are not known because Shuffle Entertainment published them, they are known because any mathematician/programmer can calculate these numbers using a variety of techniques.  In the case of Ante/Play, there are a total of 6 cards dealt to the Player and the Dealer (3 each).  There are 22,100 possible Player hands and 18,424 possible Dealer hands for each of the Player hands.  Thanks to the speed of today's computers, a program can run through ALL of these hands (numbering well into the Trillions), determine the right strategy for each Player hand and tell us absolutely everything we would ever want to know about the game - the payback, the win fequency, the probability of winning given any Player hand, how often the Player folds, how often the Dealer doesn't qualify, etc... 

            Unlike food, casino games are, well, gambling.  Part of gambling is rewarding those people who are more prepared and more knowledgeable about gambling.   The strategy for Three Card Poker doesn't take a PhD to learn.  It takes about 20 seconds (or less).  Play a Q-6-4 or better.    You could read this just about anywhere on the internet.  If you want to know the details about the strategy (how and why), you can read a booklet on the game (I suggest my very own Expert Strategy for Three Card Poker, but that's just me!)  Armed with this strategy you are very likely to do better than someone who has no idea what to do over the short run and almost assuredly so over the long run. 

            For years, people have asked me if I'm banned from casinos because of my in-depth knowledge of table games or if I'm 'hated' by the casinos for arming people with the strategies for how to play the games.  I'd like to think that I've had at least some influence on how people play.  But, I don't think the casinos care one bit.  Even with the best strategies, all casino games (except a few video poker variants) have house advantages.  The casinos are totally fine with a few people playing near the theoretical payback as results are all relative.  A few people who are winning in the short run or who are doing better than the rest can keep the rest of the Players in the game.  After all, if the other guy can win, why can't I?  Of course, this is far more true IF you know the right strategy. 

            I've often surmised that if I were to stand near a Three Card Poker table, handing out my booklet for free that only 20-25% of the Players would actually use the strategy.  Half of these people would probably give it up within the hour when the results don't match the theory - ignoring the time factor that is required for this to happen.  What this really translates to is that I don't think it would matter one bit if the casinos were to put a little sign on each table that had the payback and win frequency of the game.  Most Players would probably ignore them.  After all, how many people really read those nutritional labels on food anyhow?  And that's about what actually goes INTO your body!