The term expected value, or EV for short, appears frequently in articles and game strategies. Video poker is a game of pure mathematics, making it possible to precisely determine the EV for every play. The concept of EV is cardinal to a successful strategy for any game. So just what is meant by EV?

To put it in as simple terms as possible, the EV of any chance event is the average of all possible outcomes. When most people hear the word average they usually think of what mathematicians call the mean; that is, you add up a bunch of numbers and divide by the number of numbers. To calculate the EV, however, we must use a weighted average, not just the mean.

For example, suppose you are playing common 9/6 jacks or better video poker, and you hold an ace and a jack of the same suit. The possible payoffs are a royal flush, which pays 800-for-1, four of a kind at 25-for-1, full house at 9-for-1, flush at 6-for-1, straight at 4-for-1, three of a kind at 3-for-1, two pair at 2-for-1, a high pair at 1-for-1, and zilch (no payoff) at 0-for-1.

The mean of these possible payoffs, which is (800+25+9+6+4+3+2+1+0)/9 = 94.44, tells us absolutely nothing of any useful value. To get the EV, we must take into account the probability of each payoff, as I show below.

If you multiply the amount of your bet by the EV of the play, the result is the expected return, which is the average that you can expect to get back if you made that same play many times. For example, an EV of 1.023 would mean that you have a 2.3% advantage, and you can expect to get back an average of $10.23 for each $10.00 bet. Conversely, an EV of 0.982 would mean that you could expect to get back an average of only $9.82 for each $10.00 bet, which translates to a 1.8% disadvantage. Obviously, we would like to place bets only on events with an EV greater than one.

Now let’s apply this to video poker. For any dealt five-card hand there are thirty two possible ways to play it, and for each of these thirty two ways of playing a hand there may be many possible outcomes. Assuming an honest video poker game, each unseen card has equal probability of appearing at any time. Assuming that you’re playing a consistent strategy, this makes it possible to calculate the probability of each possible payoff. The EV of any play can then be calculated by multiplying the probability of each possible outcome by its payoff and summing the products.

Another way to look at it is to multiply the number of ways each final hand can occur by its payoff, sum those products, and divide by the total number of possible outcomes. Using that method, let’s determine the EV of one video poker play.

Suppose we’re playing 9/6 jacks or better, and we want to know the best play when dealt: Kh Qs Jh 10h 4h.

Draw 1, hold K-Q-J-10, hoping for a straight. We are drawing one card from the remaining 47 unseen cards, so there are 47 equally likely possible outcomes, as follows. Any ace or nine will make a straight, which pays 4-for-1, and there are eight such cards in the deck. Also, any king, queen or jack will make a high pair, which pays 1-for-1, and there are a total of nine such cards remaining in the deck. Any of the other 30 remaining cards will yield zero payoff, so the expected value can be computed as:

(8×4+9 x1+30×0)/47 = 0.8723

Of course, the “30 x 0” term doesn’t contribute anything to the EV, it is purely for demonstration purposes.

Draw 2: Hold K-J-10-4 (all hearts), hoping for a flush. Any heart will make a flush, which pays 6-for-1, and there are nine such cards remaining in the deck. Also, any king or jack will make a high pair, which pays 1-for-1, and there are six such cards available. None of the other 32 cards yield a payoff, so this play has a total expected value of: (9×6+6×1)/47 = 1.2766

Draw 3: Hold K-J-10 (all hearts), hoping for a royal flush. This is more complicated because we are drawing two cards instead of one. Since there are (47*46)/2 = 1,081 possible outcomes instead of just 47, I’ll skip the mathematics and tell you that the expected value is 1.3506.

Therefore, if you were dealt this same hand many times and always drew for the straight, you could expect an average return of only a little over 87 cents for each dollar bet, so you would be losing money on the play. However, if you always drew for the flush you could expect an average return of $1.27, and if you always drew for the royal you could expect an average return of $1.35 for each dollar bet. Which way would you play the hand?

Note that in each case we are usually hoping for one particular final hand, which I call the primary payoff, but part of the expected value (sometimes most of it) comes from other possible outcomes, which I call secondary payoffs. In draw number three above, for example, nearly half of the expected value comes from secondary payoffs (i.e. final hands other than a royal flush). This is good because it reduces the volatility of the game. That is, it reduces our likely bankroll fluctuations.

There are 29 other ways to play this hand, but these three give the highest expected values. If we were building a hand rank table, we now know that a three-card royal flush (abbreviated RF 3) is generally higher than a four card flush (flush 4), which in turn is higher than an open-ended four card straight draw (straight 4).

By analyzing hundreds of different hands in this way, we can build a complete hand rank table, which can then be used as a playing strategy. Once you have an accurate hand rank table, it is no longer necessary to consider specific expected values. It is the job of analysts like myself to provide such tables, so that anyone can play optimum strategy without having to understand the math. For any given dealt hand, you simply scan down the table and hold the first available combination.

I am sure that most video poker players are familiar with the expressions ‘full pay,’ ’short pay’ and ‘partial pay’, but, others may ask what these mean. The explanation given below should be required reading before play on any VP machine: reading the pay schedule. The best place to start is with a basic ‘plain vanilla’ vp machine pay schedule for jacks or better (jacks+). This version is usually the choice of newbie VP players because of its simplicity, however many VP veterans nominate Jacks or Better as the best VP game. Following is a full-pay 9/6 regular VP pay schedule for jacks or better: (Multiply by 5 for maximum coin play.)

strategy tips for jacks+:

• Always keep a five-card winning ‘pat’ hand unless you can draw one card to a royal flush, or a straight flush
• Break a high pair only for a 4-card royal or any 4-card straight flush including an inside draw
• Break a low pair for any 3-card royal flush or any 4-card flush or straight flush
• Break a 4-card flush or open-ended straight for a 3-card royal
• If you have both a 4-flush and 4-straight, go for the flush, the better paying hand.
• An inside straight is completed with one card only, (e.g. dealt 6, 7, 9, 10). Here only the eight can produce a winning hand. This is a good bet to complete a straight flush or royal.
• An outside straight is completed with two possible cards, (e.g. dealt 6, 7, 8, 9). Here a ten or five can produce a winning hand. This hand should always be held for a possible win.
• Ace is not the most important card, especially if not suited with another high card. The jack is, because you have more opportunities to develop winning hands around a jack than around an ace. To form a sequence, a jack has two possibilities: a queen on one side and a 10 on the other. The ace however, has only one possible sequence partner: the king. In addition, when dealt two face cards and an ace, not suited, keep the two face cards only and discard the ace allowing three opportunities, rather than two, to either match the face card(s) or form a straight.
• Never keep a kicker. That’s an additional face card with any pair; this reduces your return by 5%. Hold only the pair and draw three cards. Approximately 1/3 of all hands dealt will have no value other than high card(s) and 20% will be paying hands.

Jacks+ 9/6 strategy is easy to learn and that is the beauty of this VP version. Note: These strategy tips are for Jacks+ 9/6 version only. All other VP versions with varying pay tables require specific strategies for each. I have included a Jacks+ Strategy Card below for your convenience.

Jacks+ VP strategy card draw to the hand higher on the list high card is a jack or better.

Keep in mind that in a grouping of video poker machines, known as banks or carousels, some casinos will ‘mix-in’ several VP machines with inferior pay programs (short/partial pay). This means that the educated player, on the full-pay Jacks+ 9/6 machine, will be winning more for the same hands than the other players will.

Many strategists say optimum play when they really mean perfect play; that is, every play would be made for the highest possible expected value, even if the cost of a simplified play is miniscule. For all attractive video poker games, only a computer would be able to make the absolute best play in every situation.

For some games, we can come very close to the maximum payback with just a few strategy rules. For others, it requires a hand rank table as shown elsewhere in this site for Double Bonus Poker at about 100.15% and All American Poker at about 100.7% payback. Such games seduce some strategists into generating a very detailed strategy with lots of penalty card situations (penalty cards will also be discussed in an upcoming article).

Some “experts” are so concerned about perfection that they generate penalty card situations ad nauseam. I do play video poker quite a bit, and I could learn those complex rules if I wanted to, but it would be a wasted effort. With my strategies I can come within a couple of hundredths of a percent of perfect play, and I can play about 500 to 700 hands per hour. Adding the complexity of all those penalty cards would slow me down, cutting into my expected win rate much more than the potential gain of the better strategy and greatly reducing the fun of playing. For any endeavor, there is a point of diminishing returns.

In the case of a complex video poker game such as All American Poker, you can use a very simplified strategy and play very fast, but you will be giving up a significant part of your potential win rate. Learning a somewhat more accurate strategy will increase your expected per-play gain with little loss of speed, resulting in an increase in your hourly win rate.

A strategy might be developed that would yield very close to the maximum payback, but it would necessarily by quite complicated. Following such a strategy faithfully would increase your per-play expected gain, but considering those penalty card situations would probably slow you down.

Worse, you might make more errors through inadvertent deviations from the strategy and thus have a lower net win rate. Because of the wide difference in the various games and various people, I feel that the term optimum play should be defined as ‘the strategy that will yield the highest per-hour expected win rate for a particular player.’ Note, especially the reference to a particular player.

I don’t know anything about your ability and dedication to be a winning player, so I don’t know exactly what your optimum strategy would be, so let’s examine what I do know. I know that the industry-wide average hold on all video poker games in Nevada is about two percent more than it would be if everyone played perfectly, and the reason is that the vast majority of players play by guesses and hunches instead of learning a good strategy.

Is this due to lack of ability? No, nearly everyone has the ability to learn to be a winner, but most either don’t try because they are in a casino just for fun, knowing that they will most likely loose the money they have brought along to gamble, or they believe that it doesn’t matter how they play because the machines are set to a specific payback.

Contrary to common myth, the machines are random, and in most jurisdictions they are honest, so you can be a winner if you want to. In the previous column I presented simplified versions of my Precision Play rules for Jacks-or-Better Draw Poker and Deuces Wild. While these greatly simplified strategies obviously will not yield the full potential payback of the games, they do come within one or two tenths of one percent.

For games with widely ranging payoffs and probabilities, such as video poker, it has been very difficult to determine bankroll requirements. Until recently, we relied mostly on computer simulations for an approximation, but Russian mathematician Evgeny Sorokin has shown that the risk of ruin (the probability of losing one’s entire starting bankroll) can be calculated precisely as a root of equation 1 below.

The formula may look complicated, but don’t let that stop you from reading this article, even if you’re not mathematically inclined. As a serious player, you will find this information very valuable. For several of the most attractive games, you can skip the math altogether by using a table published in Video Poker Times. Take a few minutes to look at the equation.

Simply stated, the risk of losing a one-bet bankroll is equal to the summation of a function of all the possible payoffs and their probabilities. Although the rebate is not actually paid immediately on each play as implied by adding C to W, the result is very close, since you can go to the slot club booth and make the cash any time it reaches the club’s minimum transaction level. As a starting point, this formula assumes that your entire gambling bankroll consists of exactly one betting unit (five coins for a typical video poker game), but it’s easy to extend that to a real bankroll. You place your one bet, and if you lose, you’re done. If you win or get a push, you continue playing indefinitely until you either lose that starting bet and all winnings, or have amassed a fortune.

Let’s apply this to a specific game. Suppose you play full pay double bonus poker very accurately in a casino that pays 0.25% cash rebate on the slot card, and you want to know your risk of ruin (RoR). The per-coin payoff (assuming five coins played) and the probability of each final hand type (assuming perfect play) are shown in the double bonus poker strategy section, and C will be 0.0025 for all cases. The Zilch (no payoff) entry is never shown on the machine’s payoff schedule, but its probability is given by the game analysis program, and it must be included in this calculation as the i=0 case.

Since R(1) is the summation of a series involving a function of itself, it can be found only by an iterative procedure; that is, make a guess, try it, use the result as the new guess, and repeat until the result is as accurate as desired. The series will converge faster if we start with a reasonable guess such as 0.99 and then average each result with the previous guess for use as the next guess. Using final hand probabilities and payoffs for standard full pay double bonus poker plus 0.25% slot club rebate, we find that the risk of losing a one-unit bankroll is R(1) = 0.99970164. (With real human play, the risk is a bit higher, typically in the fifth or sixth decimal place with skillful play)

Subtracting from one, we see that the probability of building a single 5-coin bet into a vast fortune with perfect play on this game is 0.0002984, or about one chance in 3,350. That may not seem like a very good prospect, yet it’s a lot better than any state lottery. Also, your starting bankroll is presumably at least a little bigger than just one 5-coin bet. What we really want to know is the risk of losing that real gambling bankroll. The probability of losing an entire starting bankroll of B betting units is given by equation 2. Since R(1) is less than one for a positive expectation game, a bigger starting bankroll results in an exponential reduction in the risk as follows:

For example, if you start out with 1,000 betting units (e.g., $5,000 on a 5-coin $1 double bonus machine) and play at a casino with a 0.25% cash rebate slot club until you either lose it all or amass a vast fortune, you have almost a 26% chance of becoming very rich (although at an expected average win rate of only 0.4% of your action, it’s going to take a very long time).

As a more practical concern, you probably want to know how big a starting bankroll you need for a specific risk of ruin that you consider acceptable. For example, suppose you decide that you can live with one chance in ten of losing your entire bankroll. You could interpolate from the above table and estimate that you need about 8,000 betting units for RoR = 0.1, but why not calculate it exactly? Solving equation 2 for B, we get equation 3. Plugging in 0.1 for R(B) and 0.99970164 as determined above for R(1) we get B = 7,715 betting units. Multiplying by $1.25, we find that we need to start with at least $9,644 to limit our RoR on a quarter Double Bonus game to ten percent.

Now let’s look at another game. For standard full pay deuces wild, R(1) computes to 0.9993469 (with no slot club rebate). This doesn’t seem much different from the figure for double bonus poker, but the third column above shows what happens when we raise it to large powers of B to compute RoR. Note how much lower the risk is with any reasonable bankroll. Combine that with up to five times the average win rate and a much more straightforward strategy, and you see why I recommend deuces wild for beginners in Las Vegas. On a quarter deuces wild we would need only $4,406 for that same 10% RoR, and that is even without a slot club rebate!

As you can see, the risk of ruin when playing video poker can now be determined precisely. You can do the calculation yourself, or Video Poker Times issue 7.5 presents a table giving the necessary bankroll for whatever you consider to be an acceptable risk of ruin for a dozen attractive games. (Note: There is a typographical error in the cash rebate for Sam’s Town in that table. It should say 0.093%. The RoR numbers for bonus deuces are correct for that rebate.)

There are always two roots of Sorokin’s equation; one, and the result we are seeking, which we are hoping is less than one. (If the root is greater than one then the game offers less than 100% total payback, and ruin is certain if you play long enough.)

equation 1:
equation 2:
equation 3:

Key: R(1) is the risk of ruin with a one-unit starting bankroll, n is the number of entries in the game’s payoff table, i identifies one entry in the payoff table, P is the probability of final hand type number i, W is the per-unit-bet win (payoff) for hand type i, C is the cash rebate from the slot club as a decimal fraction, B is the number of betting units in your starting bankroll, and R(B) is the risk of ruin with a starting bankroll of B bets. Note: One betting unit is typically five coins for video poker.

We know that a video poker player generally can hold the casino edge to less than one half of one percent, and in many cases we can even turn it into a player advantage with only a modicum of effort. But can we still get the comps and cash rebates even if we are playing with the advantage?

Note what I said above; the comp system is based not upon today’s actual losses (or winnings) but upon the casino’s perception of your expected losses. Since the vast majority of video poker players play at about two percent less than maximum payback, the casino generally sees video poker players as losers just like all other gamblers. (Otherwise, why would they have the machines in the first place?) Therefore, it is usually possible to collect comps and other benefits even if you are a winner.

Most casinos now reward slot and video players by means of a slot club. Membership is free, and when you join they issue you a card. Although the card appears very similar to a credit card, its use is quite different. Before you start to play, you insert the card into a reader on the machine. This identifies you so that the computer can track your play. In some casinos they only keep track of your action (total money wagered), but most also record your wins and losses.

The rewards may come in one or more forms, as follows:

• cash rebates: Usually figured as a fixed percentage of your action, typically ranging from 0.05% to 1%. Many casinos do not offer cash rebates.
• comps: Most often complimentary meals or accommodation. These may be in addition to or in lieu of cash rebates.
• intangibles: Actually another form of comp, but may include show tickets, use of the limo, etc.
• mailings: Joining the slot club puts you on their mailing list, and most casinos mail out monthly flyers. Often they include vouchers for discounts at restaurants, match plays, or even free money when you buy in. Often these vouchers are related to your level of action. They usually keep track of your action in the form of points. The most straightforward clubs award one point per dollar played. Others want to claim that their points are worth more. Of course they do this by requiring more action for each point earned, so the advertising can be very deceptive. The important thing for you is to figure out the actual total return as a percentage of your action.

Let’s look at a rather broad example. Suppose the casino gives one point per nine quarters played. Thus, each point requires $2.25 of action. When you have accumulated enough points you can get:

• $5.00 cash for each 1,000 points redeemed
• a buffet (regular $8.95 plus .65 tax) for each 1,250 points earned
• a show ticket (regular price $49.95 plus 3.62 tax) for each 5,000 points earned.

Note that you have to redeem points for the cash rebate, but in this club the other rewards are granted for points earned without having to cash your points in. To make the calculation easy, let’s assume you have earned 10,000 points. Doing this required 10,000 x $2.25 = $22,500.00 action. On a quarter video poker this would require $22,500.00/$1.25 = 18,000 plays, or about 30 hours of play for most players. For that 10,000 points, you can get 10,000/1,000 x $5.00 = $50.00 cash. Dividing by the $22,500 action indicates a 0.222% cash rebate.

That’s not bad by itself, but remember that you also get 10,000/1,250 x $9.60 = $76.80 worth of buffets, plus two show tickets worth $107.14, for a total return of $233.94 in values. Assuming that those comps have full value to you, that’s a total of 1.04% returned to you. Even if the best game available offers only a little over 99% payback, the potential total expected return is over 100%. The slot club has turned a negative situation into a positive one.

How can the casino return this much?
First of all, remember that most players are going to get less than 98% return from the game, so the casino still has its edge. But more importantly, note that the biggest part of the rewards are what are called soft comps. The buffet probably costs them quite a bit less than $8.95, and it keeps players coming back into the casino to eat those ‘free’ meals and play some more. The show tickets may be for nights that the show is never sold out, to their actual cost is negligible.

What does this return mean to you?
Suppose you’re playing full pay Jacks or Better at about 99.52% return. Adding the slot club benefits, you find that you have a positive expectation at over 100.5% total payback. Not a big profit situation, but what if the casino has a double points day? Doubling the slot club benefits boosts your total payback nearly 101.6%. Of course, that’s assuming you can use all those buffets and show tickets. Well, if you can’t use them all yourself, why not treat your friends or business associates to a meal or show?

The point is that the slot club rewards can make up a significant portion of your expectation. When choosing a casino, try to weigh all the factors. Advertising can alert you to promotions and changes in the slot clubs, but now you know how to evaluate them for yourself. Doing so is an important step in choosing a casino.

If you want a get rich quick scheme video poker is not your game. If you are looking for a way to overcome the house edge that is built in to nearly all casino games, again you will be disappointed. If you want to know how to find a hot machine, you will be depressed to know that there is no such thing. But if you want to learn how to achieve a small but real advantage on certain video poker games (and have fun doing it) then read on.

My publications show you how to keep the house edge to a minimum, and, in some casinos, to even gain a real mathematical advantage over the house. My strategies are computer-generated and proven by professional players, yet they are easy for a novice to begin using immediately. You can become a favourite to be a winner in the long run and have fun at the same time.

For more than two decades, casino visitors have been intrigued by video poker. They may have heard that the payback is higher than reel slots (which is generally true), perhaps they like the challenge of making decisions instead of just pulling a handle, or maybe it’s just to try something new. After a short try, however, many move back to the one armed bandits. Why do they give up so quickly? Is video poker too difficult? Are they losing even more than they did on the reel slots? Is it the five coin requirement to qualify for the royal flush jackpot? Or is it just that they have a very small budget for gambling and there are too few nickel video poker machines available?

This article delves into the whys and wherefores of video poker for the novice, especially one whose gambling has been limited mostly to nickel or quarter slots. By seeking out the good machines and following a few simple playing rules, you can cut the house edge to one half of one percent or less, and in some cases you can even gain an edge over the casino, even on some nickel machines.

Already I hear you say, “I don’t want to be bothered with rules. I’m on holiday, and just want to have some fun.” But isn’t it more fun if your limited bankroll lasts longer, and even more fun when you’re winning? The big advantage of video poker over the no brainer reel slots is that the maximum payback can be determined from the payoff schedule on the front of every machine. No, you don’t have to do any maths; just compare the payoff schedule with my tables. Most players’ intuitive decisions will yield about 1% to 5% less than a game’s maximum payback. Statistics from Nevada casinos show an average payback on video poker of about 2% less than what it would be if everyone played perfectly.

This is why the casinos are able to offer games with over 100% payback potential, thus making it possible for us to gain a small but real edge on some games. Most video poker experts assume computer-perfect play when rating a game’s payback. Such accurate play is impossible for most of us, yet you can easily get within a few hundredths of one percent of the rated payback with my simplified strategies. Even in a casino that doesn’t offer 100% plus games, video poker generally offers a higher return than any other low stakes game.

In this article, we look at how to pit your wits against the machine using purely brain power; once engaged, you have a 50-50 chance to overcome the house advantage.

It’s a common tip that to play a video poker game without the advantage is in the vicinity of insane. But what does that imply? That there are people out there taking advantage of some stupid mistakes by the casinos? Yeah right, and a blackjack pays 5 to 1.

In reality, (and possibly new to you since noone cuts through the bull) – nobody ever walks into a casino anywhere, sits down at full pay deuces wild, 10/7 DB poker, or 9/6 JB with creative comps, and plays with a mathematical advantage of any kind. Why? Simply because those advantages are theoretical over an infinite amount of time with absolutely unattainable computer-perfect play, and even the most addicted players pound away less than 8 hours a day, which is nothing more than a disconnected series of short-term attempts.

The casinos know this – why don’t they? The math never lies, but playing to the math is impossible. Where this all started that the casinos would actually allow people to play their machines, who so blatantly announced their superiority over certain games with any kind of credibility, I’ll never know. Name one casino in Nevada that worries about such stuff. They certainly allow them to play all right – to the tune of taking their money.

Now let’s use our heads some more. Who said the human mind is limited to never figuring out ways to overcome odds of any kind? Based on our history we already know this is untrue. When it comes to games of chance, the mathematicians always seem to have the last word. Not any more. Up until 4 years ago, while I played optimal strategy, I had less than a 15% rate of leaving casinos a winner. Since then I’ve had a greater than 98% success rate, and it is dead on target to the expectation that took me nearly 3 years to develop with the help of several consultants in many countries. It is loaded with special manoeuvers and inconsistencies, but is simple in its premise of setting pre-determined win and loss goals, and never deviating from them.

Progressive in nature – after losses and wins – it is far from the dangerous Martingale system, which plays a minor role in the overall methodology. Willpower, determination, bankroll, and the ability to always do what you say you are going to do – a feat almost impossible to perform while involved in casino action – are the ultimate keys. It is the true test of a gambler who has entered the arena for the purpose of gambling to win, and not simply to have his performance rated.