Surrender and Insurance

            As I’ve described many times, the concepts of Expert Strategy apply to more than just video poker.  Essentially, they apply to every game in the casino (except slots of course).  You should always know which games to play, what strategy to play them with and what to expect.  Most games in the casino do not require learning very significant strategies to play them properly.  Two that do – video poker and blackjack (and its variants Blackjack Switch and Spanish 21 – require some serious effort to learn them correctly.  The reward for doing so is a payback that is above 99.5%.

            In order to achieve the theoretical payback, you have to learn ALL of the strategy including the less well-known parts and even the parts we might find ‘offensive.’  For blackjack this would be the concepts of insurance and surrender.  The idea of ‘surrender’ is the one that you may find to be ‘offensive’, but there are times it is the right play.

            First let’s begin with the definition of the Insurance bet.  When the Dealer has an Ace up, he will offer everyone at the table the option to make an Insurance wager (which must be ½ of your base blackjack wager).  In reality, it is nothing more than a proposition bet.  If the Dealer has a blackjack, then you win 2 to 1.  If he doesn’t you lose your Insurance wager.  Assuming you have not been counting cards, then the odds of the Dealer having Blackjack is roughly 4 out of 13 (I’m ignoring his upcard ‘Ace’ and any of the cards you can see).  Paying 2 to 1, gets us back 12 out of every 13 units wagered for a payback of about 92.31%.  Obviously, you can do some light card counting and only make this wager when it is more in your favor, but it will take a lot of non-10s/Faces to turn the deck in your favor.

            Sometimes you will hear a Player who has a Blackjack to ask for ‘even money’ when the Dealer has a Blackjack.  This is really the equivalent of the Player making the Insurance wager.  If he makes it and the Dealer does NOT have Blackjack he will win 3 for 2 on his base wager, but would have lost 1/2 unit on Insurance leaving him having won even money.  If the Dealer DOES have blackjack, he pushes his blackjack wager and wins his Insurance wager, which will pay 2 to 1 of the INSURANCE wager which is equal to his base wager – in other words, even money on the base wager.  To keep things moving along, most casinos will just allow the Player to call “even money” and get paid 1 to 1 on his blackjack wager. 

            In reality, this is no better a decision than making the Insurance wager under any other situation.  However, from an emotional standpoint, many Players hate the idea of a total push when getting a Blackjack.  This would be the outcome if you don’t take the Insurance Wager AND the Dealer has Blackjack.  The proper play is to stay unemotional and never take even money.  This situation should only occur about 1 in 275 hands (approximately) which would mean once every 9 hours of play.  For some strange reason, I seem to get it about 3 times an hour?!

            Next up is the Surrender rule.  Many of you may never have heard of it.  The casinos don’t really advertise it much.  You have the right to Surrender your hand before you take any other action by forfeiting half of your initial wager.  Once you hit, split, double down, etc… you can no longer Surrender your hand.  There are two different variations of Surrenders.  The first called Early Surrender is rarely offered.  It allows you to Surrender BEFORE the Dealer checks for a Blackjack when he has a 10/Face or an Ace up.  Thus, even if the Dealer has a Blackjack, you would have forfeited only half of your wager.  This is a big advantage to the Player which explains its rarity.  The other variation is called Late Surrender.  This version has the Dealer checking for Blackjack and only after it is confirmed that he does NOT have one can the Player opt to Surrender.

            Unlike the Insurance Wager, this is not a proposition wager better left ignored.  If that were the case, the casino would have it on the felt in big bold letters “PLEASE SURRENDER!”  Instead it is an option you need to take on occasion and you almost have to ask the casino permission to do so.  From a mathematical perspective, the decision is easy.  If you are going to win less than 25% of the time with your starting two cards, you Surrender.  At a 25%, you would win back exactly half of your initial wager which is what you’ll have left after Surrendering.  Hence, that is why this is the decision point.  There are slightly different strategies depending on whether the Dealer hits or sticks on  Soft 17. 

            You should always Surrender a Hard 16 to a Dealer 9, 10 or Ace.  You should also always Surrender a Hard 15 to a Dealer 10.  If the Dealer hits a Soft 17, you also Surrender a Hard 15 to a Dealer Ace and a Hard 17 (yes, I said 17) to a Dealer Ace.  If the Dealer has a 6 underneath, he gets to keep going and is that much more likely to wind up beating you.   These rules apply to larger shoes of 4-8 decks. 

            The impact of properly Surrendering is that the payback is increased by 0.07%.  This may not sound like a lot, but looked at differently, it can cut the house edge by about 15%.

Is the Search for Perfection Overrated?

            

            One of the traits I inherited from my father, Lenny Frome, is that I am a perfectionist.  This is not to say that I am perfect (far from it).  The joke in our family was always that one of my sister's brought home a '99' on a test and my father, not missing a beat said "why not a 100?"  Fortunately, for me, by the time I came along, he learned to temper his ways a bit.  I try to focus most of my perfectionism inward.  Nobody takes it harder when I find a mistake in my work more than I do.

            At the same time, I try very hard to be practical about things too, where strategy is concerned.  Us mere mortals do have limitations to our ability to memorize dozens of video poker game strategies.  It is why I strongly recommend that you learn one or two different games and do your very best to 'perfect' the strategies to those games.  But, what are you to do if the games whose strategy you memorized are not available when you go to the casino or if you're just in the mood to try something different?

            In these cases, you just have to use some common sense.  If you try to bring your Jacks or Better strategy over to Joker Poker, you may find yourself in deep trouble.  But, what happens if you use your basic full-pay Jacks or Better strategy on a full-pay Bonus Poker machine?  What will this really cost you in theory?  Calculating this - with the help of some of my video poker analysis programs - is relatively easy.

            We simply run the numbers on a full-pay jacks or better machine using Expert Strategy for jacks or better.  From this program, we extract the frequencies of all of our winning hands.  We then use these frequencies against the Bonus Poker paytable to get a theoretical payback of Bonus Poker using jacks or better strategy.  We then run the numbers on Bonus Poker using Expert Strategy for Bonus Poker and compare the results.

            The theoretical payback for Bonus Poker using Expert Strategy for Bonus Poker is 99.16%.  The theoretical payback for Bonus Poker using Expert Strategy for jacks or better is 99.15%.  If you were to play all 2,598,960 possible 5-card deals using jacks or better strategy you would find that you cost yourself about 200 coins.

            In other words, while it is still preferable to learn the right strategy for Bonus Poker, if you use your jacks or better strategy, your bankroll will not take a big hit.  However, this should not give you free license to play jacks or better strategy on any game you want.  If you were take your jacks or better strategy to a DOUBLE Bonus Poker game, you would find that it will play at about 99.6%.  This might sound good (after all jacks or better itself plays at a bit less than this), but you have to remember that Double Bonus Poker is one of the few positive games out there.  If you play it using the proper strategy, it can afford you a 100.1% payback.

            So, what's the point?  Good question.  Nobody should expect perfection when they head to the casino.  The simple mathematical fact is that every deviation from perfection, however, will cost you.  It might be peanuts as in the case of playing Bonus Poker using jacks or better strategy.  If you play a hundred hours a year as a $1 max-coin Player, you'll cost yourself $30 per year.  We'd all rather have that $30, but we're not talking a significant amount of money.  Do the same on our Double Bonus scenario and we're talking about $1500 per year, which I dare say is QUITE significant.

            So, while we shouldn't shy away from video poker because we might make a few mistakes, we should still be prepared to learn the right strategy for each game and to do our very best to utilize it every time we play.  This is the very essence of Expert Strategy - Know which games to play, know what strategy to use and know what to expect.

            For those of you who want to learn the subtle differences between jacks or better strategy and Bonus Poker strategy, both strategies can be found in Expert Video Poker for Las Vegas and Winning Strategies for VideoPoker.

Get Up to SPEED - Let It Ride and Mississippi Stud

            

            Comparing Let It Ride to Mississippi Stud gives us a great opportunity to understand how a subtle difference in betting structure can greatly alter the strategy of the game and thus radically change a game that is otherwise rather similar.  The subtle difference in this case is that in Let It Ride the '1' and '2' wagers are completely optional (essentially, they can be 'checked') and in Mississippi Stud the choice is to Play or Fold.  No checking allowed.

            To best compare these games we need to realize that the idea that the you can take your wager down in Let It Ride doesn't change the game.  The rules of the game could have simply made the '1' and '2' wagers simple optional wagers.  You can either 'check' or you can make the wager. 

            Mississippi Stud also differs in that your first decision is after seeing only 2 cards instead of the 3 as in Let It Ride.  Mississippi Stud's paytable also goes down to a Pair of 6's, whereas Let It Ride goes to a Pair of 10's.   After 2 cards, the Mississippi Stud Player must decide whether to make at least an additional 1-unit wager or to Fold.  Mathematically, this is vastly different than the decision to check or Play.  When we have the decision to check or Play the question becomes one of whether or not the Player will win more than he loses on THAT specific wager.  Prior and future wagers play no part in the equation.  When the choice is to Play or Fold, the question becomes one of whether we can win back at least the amount we are about to wager when we consider ALL other wagers - both those already made and those we might make during the hand.  This is because if we choose NOT to Play, we are forfeiting all past wagers and the right to make all future wagers.

            The impact to this becomes most evident when we compare the '1' wager in Let It Ride to the 4th street Wager in Misssissippi Stud.  In this case, both hands consists of 3 cards and we are deciding whether to/how much to wager on the 4th card.  In Let It Ride, we find ourselves making the wager very infrequently.  We are willing to leave the wager in place only on sure winners (Pair 10's or Better or Trips), 3-Card Royals and 3-Card Straight Flushes (Open or Inside, NOT Double Inside).   We make this wager only 7% of the time.

            In Mississippi Stud, by the time we get to 3 cards, we have already wagered our Ante and at least 1 unit on the 3rd Street Wager.  If we Fold, we are forfeiting both of these  wagers.  We will also end our hand right then and there.  So, we also forfeit the right to potentially benefit from our next wager (5th Street).  The decision to Play 3x is similar to our Let It Ride decision.  Once you are going to win more than you are going to lose on a specific wager, you wager as much as the house lets you.  So, we find that we wager 3x on all sure winners, 3-Card Royals and a variety of 3-Card Straight Flushes.  We still go ahead and wager 1x on a whole lot of hands that sound like they're going to need some help to become winners.  This includes all Low Pairs, 3-Card Flushes and hands with the right combination of High and Medium cards.

            The net result is that we very rarely fold at this decision point.  The overall fold rate for Mississippi Stud is 44%.   31% (or nearly 75% of the total folds) occur after you see the first 2 cards.  Of the remaining 69% of hands that go to 3 cards, you will fold only 12% of the time.

            In Let It Ride, you will let the '2' wager stay up 16% of the time.  In Mississippi Stud, you will make a wager at 5th Street more than 90% of the hands that go that far.  This happens for two major reasons.  The weakest hands were folded very early on.  A hand that started as two Low Cards was dropped early, which makes weaker hands that much less frequent later on.  In Let It Ride, even the weakest hands have a chance to make it to the end of the hand.  The second reason is that when you have 3 units already wagered and you are compelled to either Fold or make another 1-unit wager, it does NOT take a high win frequency to make it worthwhile to make that additional 1-unit wager.   With just 1 High card and 2 Medium Cards or 2 High Cards in hand, it still pays to make this wager.

            With 2 High Cards, the Player still has 6 chances to draw a High Pair which will return 8 units the Player (each).  With 48 cards remaining in the deck this amounts to an expected value of exactly 1.0, which is the cutoff for determining whether or not to make the wager.  Throw in a Medium card as well and he gets 3 more chances to pick up 4 units and the expected value is now 1.25.  If this were a check or Play decision as in Let It Ride, the decision would clearly be to pull it back with these types of hands.

            There is a reason why I've coined Mississippi Stud to be Let It Ride on SPEED.  The games are very similar in how they play but vastly different in strategy and size of bankroll needed to sit and play.  I can't quite cover all the differences or all the strategy here, but for a limited time, I'm offering up a buy one get one special on my two booklets for these games.  Buy Expert Strategy for Mississippi Stud for $5.95 and get Expert Strategy for Let It Ride for free.  Just send check or money order to Gambatria, P.O. Box 36474, Las Vegas, NV 89133 and I'll get them both out to you ASAP.

Smackdown! Video Poker vs. Slot Machines

            

            This past week, I received an e-mail regarding last week's article about slots.  In that article I talk about how casinos can legally and do make sure that slot machines are created so that they produce a large number of near misses to make the Player feel like he almost won.   A reader wanted to know if the same is true of video poker.  He wrote:

             I'm always interested in the little things casinos do to try and influence how a player thinks.  Your article spells out an excellent example of how a slot machine display can create the illusion of coming close to a big win. I was wondering if the same thing applies to some extent to Video Poker.  If I'm holding 3 to a royal flush and my 2 new cards don't create a winning hand, but one of those 2 cards is one of the cards I need for the Royal, then I might be influenced to think that I was close to hitting a royal, even though the hand is as much of a winner as any losing hand.  Do you think that VP machines are set up this way as well?

            My answer depends on the definition of "set up".  Are video poker machines specifically programmed to have Player get more near misses than one would expect to occur randomly?  Absolutely not (in most jurisdictions).  In places like Nevada the law requires that any game that uses a real life object (like a deck of cards or die) in digital form must play as random as the real-life object.  In other words, if the game uses a deck of cards, every card must have exactly the same probability of being dealt as every other card. 

            Thus, the casino CANNOT program the video poker machine to have one of the two remaining cards for the Royal Flush be drawn just so it looks like the Player came close to winning - EVEN if it doesn't change the overall outcome of the hand.   So, if the Player is dealt a suited 10-J-Q and the two cards that are supposed to be dealt are the 8D and 5C, the machine cannot change the 8D to the suited King just so the Player comes closer, but still loses.

            What makes Video Poker so superior, in my opinion, to slots is that there is no need for the casinos (or the manufacturers) to do this.  One of the beautiful things about almost any game being played with a deck of cards is that the suspense is built into the game by the very fact that a deck of cards is being used.  True, once in a while a hand is so bad, there is no suspense, but this is infrequent.  How many times have you played a hand of video poker where the first 2 cards are a Pair or 2 cards of a Royal Flush?  Your heart skips a beat as you begin to be believe you're about to be dealt Four of a Kind or maybe a Royal.  That suspense turns to much when the final 3 cards are a mess and help your hand not at all. 

            Conversely, how many times have you been dealt very little (a single High Card) and you wind up being dealt a Flush, a Straight or even a Four of a Kind?  Nobody is forcing these hands to come out of the machine.  They occur because of the nature of the random deck of cards which generates are near misses for us.

            When we look at my reader's question about a 3-Card Royal being dealt one of the necessary cards, we find that it is not such an unusual occurrence.  For a simplistic way to approximate the likelihood of this, we simply have to know that we are going to be dealt 2 cards and we are looking for one of 2 cards to appear.  So, this is roughly equivalent to giving us 4 chances to be dealt 1 card from 47 cards in the deck.  This works out to be about 8+% of the time, hardly making it a rare occurrence.

            I think this leads to an interesting question.  Does it really matter if near misses are occurring because of the nature of a random deck of cards or if it is purposefully being programmed in by the manufacturers.  Quite frankly, by itself, I don't think so.  However, I believe what this tells us about video poker machines and slot machines is the critical part. 

            Everything about a video poker machine is the result of using a random deck of 52 cards.  So, while it is random, we also know all of the probabilities with 100% certainty and thus we can calculate a payback, determine a strategy and know what to expect over the long run.  We can look at the paytable and know everything there is to know about the machine.  We KNOW that if we see 2 machines with identical paytables, they have identical paybacks.

            With slot machines, we know NOTHING.  We can look at 2 slot machines standing side by side with identical paytables and still know absolutely nothing about either of them.  We have no idea how often winning hands will occur.  We have no idea which losing hands are programmed into it and how often it will 'tease' us with near misses.  A moment ago I gave a rough estimate of how often we can expect to get a near miss when drawing on a 3-Card Royal.  This can be calculated with absolute precision too (8.3256%).  You can't do this with a slot machine just by looking at it.

            I guess in the end it comes down to the difference between NFL Football and WWE wrestling.  I don't know who will be the next Champion, but I prefer the NFL version where it comes down to the best team and not the WWE where someone decides who should win and then puts on a good show to make it happen!

SCIENCE FICTION: BELIEVING YOU CAN WIN AT SLOTS

            When I tell people that I help develop new games for the casino industry by doing the math behind them, I'm invariably asked if I work mostly on slot machines.  Ironically, I've never worked on the math behind slot machines.  I try to explain that in my opinion slot math is amongst the easiest math in the casino. 

            Developing a casino game is really two parts.  The first is the creative half that determines the specifics of the game. The second part is the math behind the game, which can frequently cause some changes in the first part.  This dependency mostly evaporates with slot machines.  Virtually every slot machine is a clone of another game from a math perspective.

            I'm a big fan of science fiction.  So, if I wanted to invent a slot machines based on Star Trek, I merely need to come up with 20 to 30 symbols that are identified with the shows.  Maybe I use the characters (Captain Kirk, Mr. Spock, etc...) or I use the different shows (the original Star Trek and The Next Generation).  It really doesn't matter.  Most importantly, I simply have to decide what determines a winning hand.  I list out all the winning hands on a spreadsheet.  I add the amount each of these winning hands should pay.  I then determine the frequency that each of these hands occur.  I do a few simple computations and play with the numbers to get a payback to my liking and I'm done.

            Now I'd like to create a slot machine based on Star Wars.  I don't need to change any of the numbers.  I just simply need to swap out Mr. Spock for Hans Solo and Captain Kirk for Luke and I'm done.  In theory, every single slot machine could be based on a single spreadsheet of probabilities and payouts. 

            How can this be done?  because essentially, slot machines are rigged.  No, they don't know who's playing, so it's not like someone with a Player Card is going to lose and someone without one is going to win.  Nor can it tell the difference between a local and a tourist.  When I say it is 'rigged', I mean that nothing about a slot machine conforms to the notion of what you see is what you get.  When you spin the wheels, you may see more Captain Kirks than any other single symbol, but that doesn't mean the probability of lining them up is any higher. 

            In that little spreadsheet I mentioned earlier, I need to list out all the losing hands too.  The slot could simply be programmed to randomly pick a losing hand a certain percent of the time, but what fun would that be?  Instead it is programmed to give you Captain Kirk, Captain Kirk, Tribble more times than you can count.  Just for good measure there will be a Captain Kirk above or below that Tribble 50% of the time.  OH, you were SO CLOSE to winning!  In reality, you were just as far away as if the screen showed, Klingon, Romulan and Ferengi!

            To put it in more familiar terms, just because the screen showed you 7-7-orange with a 7 just below the orange doesn't mean you were any closer to winning than if it showed Orange-Plum-Banana.  If you kept drawing the fruit salad, you might get bored and leave.  But by showing you 7-7-orange, you get a false sense that you just missed.

            I would love to hear from those of you who are reading this column who continue to play slot machines.  Why do you do this?  The average slot machine in Las Vegas pays about 92.3% which makes it about the worse play in the casino.  I supposed it is fun to sit down and play a slot machine with your favorite tv show on it, but is it really worth all that you are losing?  Wouldn't it make more sense to learn to play video poker or blackjack and simply buy the complete series of your favorite show on Amazon with the money you're saving?

            Maybe Ballys and IGT should introduce video poker that is themed to tv shows and movies?  They just simply need to make the deck take on the theme of whatever show we are talking about.  Imagine Batman Joker Poker where the Joker is the actual Joker from the show.  Batman can be the King and Robin can be the Jack.  Batwoman can be the Queen!

            These characters won't change the game any, which is what happens in the slot versions too.  Maybe we can get more people to give up slots and become video poker Players if we simply put their favorite characters onto the cards? 

            As we head into 2012, I don't really care what gets you to break the slot habit, I simply implore you to do so.  In the end, I think you'll have a lot more fun and your wallet will definitely thank you!

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.

HIGH PAIR

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.

LOW PAIR

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.

4-CARD FLUSH / STRAIGHT

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!

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!