Card Values

All face cards are valued at 10, while cards 2-10 are scored according to their face value.  Aces can be worth either 1 or 11.  If a hand has an Ace it is referred to as a “soft hand” because the value can change.  Hands without Aces are called “hard hands.”  The suits of the cards have no affect on scoring.

What Wins

The highest hand in the game is a blackjack, which is an ace and a 10.  This pays the player 3 to 2 unless the dealer also has a blackjack and then the bet is pushed.  Every other winning hand pays even money and is achieved by the player having a higher total than the dealer without breaking (going over) 21.  If either the player or dealer goes over 21 they bust and lose.  If the player busts he automatically loses even if the dealer also goes over 21.  If the player and dealer have the same total the bet pushes.

Betting & Dealing

The game begins when a player makes his bet.  This is done by placing chips in the circle or logo in front of him on the table.  Only chips that are placed in the betting circle count as your bet.  The dealer will then go around the table and give each player two cards face up while dealing himself one up and one down (known as the hole card).  After the initial round of cards are dealt, the player to the dealer’s left will make the first decision on how to play his hand.  His position is called “first base.”  The action then progress around the table until all players have acted.  The last position to play is known as “third base.”

Options

All decisions are made with hand signals.  If the player wishes to stand, he waves his hand.  A player who stands is satisfied with his cards and does not wish to take any more.  If the player wishes to hit he will tap the table.  The dealer will then draw him another card.  If the player wishes to double down, he places another wager of equal value behind his original.  When you double down you will receive only one additional card.  If the player wishes to split, he places another wager of equal value next to his original bet.  You can only split with two cards of the same value.  You will then be playing two hands with one from each starting the new combination.  You can stand, hit, double down, or split these new hands.

Insurance

If the dealer has an ace in front of him he will ask the players if they would like to buy insurance.  This is a side bet where you wager half of your original bet that the dealer has 10 in the hole and thus, a blackjack.  If you make this bet and the dealer has a blackjack, you basically push your original wager.  Another form of insurance is when you have black and the dealer has an ace showing.  You can take even money instead of 3 to 2.

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Obvious No-Hit Situations

Whenever you get to 17 you do not want to hit, but taking a look at your chances of busting will give you some insight as to just why you don’t want to do so.  Here are the values in your hand, plus the chance of busting if you take another card:

20 – 92%
19 – 85%
18 – 77%
17 – 69%

Tough Decisions

To hit or to stand? These are the hands were you face the tough decisions. What you end up doing will depend on what the dealer is showing and here are your chances of busting by taking another card:

16 – 62%
15 – 58%
14 – 56%

Likely Hits

If you are holding a 13 or 12 then you are going to want to hit a majority of times. Worried about busting? Don’t be as you have a much higher chance of improving then breaking:

13 – 39%
12 – 31%

No Bust Situation

Of course it should go without saying, but if you have 11 or less then you can only improve and don’t have to worry about busting.

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Roulette - Remember that guy that sold all of his possessions and put everything he owned on one spin of the roulette wheel?  Well, this is sort of like that, sort of.  What the article advocates is what I would call a “chase” system.  Basically, you pick one color and bet it with $10, for example.  If the bet wins, great, bet $10, however, if the bet loses, double your bet to $20, if that one loses, double that bet to $40, and on and on until you win your original bet back.  The idea is that you have about a 50 percent chance of winning on each spin (slightly less if the wheel has a zero and/or double zero), and that probability says, so long as you have deep enough pockets, you’ll always eventually win your initial bet back.  I would keep in mind that most casinos have table limits to try to guard against this to an extent, but it isn’t a terrible idea, anyway.

Blackjack - According to the article, using basic card counting can increase your odds to win at blackjack by up to 5 percent.  The strategy is pretty straightforward.  Start at 0, then add 1 when you see a low card (2-6) and subtract 1 when you see a high card (10s, face cards, and aces), and don’t add or subtract anything on 7-9.  When you total gets high, you increase your bets, likewise, when you total falls to the low end, you’ll want to decrease your bets or find another table.   The idea is that you will increase your probability of receiving high cards when your total is highest, increasing your chances to win.

Lottery - This idea isn’t so much to increase your chances of winning the lottery, but to maximize your winnings if you should be so lucky as to hit the lottery.  The concept here is that you want to pick numbers that aren’t commonly used.  The number 7, for example, is a no-no because it is so common.  The article also suggests staying away from numbers between 1-31 because people often pick the numbers in their birthday for the lottery.  The purpose of this is to avoid sharing your winnings if you do win the lottery.  You’re odds aren’t very at winning to begin with, so why not make sure you are the only one getting paid if you do win?

Originially posted on BetFirms

Blackjack can be played with one to eight ordinary decks of cards. Cards of rank 2 through 10 are scored according to their face value. All face cards are 10 points. Aces are semi-wild and can be worth either 1 or 11 points. The highest hand in blackjack is an ace and any 10-point card and is called a blackjack. A winning blackjack pays 3:2. If both player and dealer have a blackjack the bet is a push. Aside from a blackjack a winning hand pays even money. The player wins if his hand has more points than the dealer, without going over 21. Thus a 21-point hand is the highest and is why the game is sometimes called 21. If either the player or dealer go over 21 it is called a break or bust and automatically loses. If both the player and dealer bust the player loses, where lies the house advantage. If the player and dealer tie the bet is a push.

A round of blackjack begins with each player placing a bet in the circle or logo directly in front of him. Then the dealer will give each player and himself two cards. Player cards are usually dealt face up. One dealer card is dealt face up (the up card) and the other face down (the hole card). If the dealer has a ten or an ace as the up card it is possible he has a blackjack in which case all player hands will lose except with another blackjack. In the U.S. the dealer will check for blackjack immediately if one is possible and will collect all losing bets immediately if he does have a blackjack.

In the event the dealer has an ace as the up card he will allow the players to insure their hands against a blackjack. This is much like any insurance policy in which you are betting something bad will happen. The insurance bet in blackjack pays 2:1 if the dealer has a blackjack. If the dealer has an ace showing and a player has a blackjack the dealer may ask “even money?” This is because if the player has a blackjack the net result of both the blackjack and the insurance bet will be an even money win regardless of whether the dealer has a blackjack. After all players have had a chance to accept or decline insurance the dealer will check the hole card.

After it has been established that the dealer does not have a blackjack the player in turn may play their hands. The following options are available.

Stand: If the player is satisfied with his hand as-is he may stand pat. To signify you wish to stand wave your hand as if to wave the dealer away. In a single deck game tuck your cards face down under your bet.

Hit: If the player wishes to take another card he may continue to do so until he either stands or busts. To signify you wish to hit tap the table with your finger. In a single deck game scrape your cards lightly against the felt.

Double: If the player feels he needs one and only one more card then he may double his bet and be dealt one more card, good or bad. This option is only offered on the first two cards, and sometimes on the first two cards after splitting. To signify you wish to double place another wager next to your original wager of equal value. In single deck place your cards face up by your bet.

Split: If the player’s first two cards are of equal point value he may split them into two hands. In this event each card is the first card of a new hand. The player must also make another wager of equal value to the first for the second hand. The player may usually resplit up to 2 or 3 times if another splitting opportunity arises. Doubling after splitting is usually but not always allowed. To signify you wish to split put the additional wager next to the original wager. In single deck place your cards face up by your bet.

Surrender: Finally some casinos offer the player the option to surrender on the first two cards. If the player does not like their prospects he may forfeit half the bet as well as his cards. This option is generally only offered after the dealer checks for blackjack, known as “late surrender.”

After all players have played their hands, from the dealer’s left to right, the dealer will play his hand. The dealer has no free will but must always play by certain house rules. Usually the rule is that the dealer must hit until he reaches a score of 17 or more. Some casinos stipulate that if the dealer has a soft 17, consisting of an ace and any number of cards totaling 6, he must also hit. If the dealer busts all players that did not bust automatically win.

The Basic Strategy

The most important thing to know about blackjack is the basic strategy. This strategy is simply the best way to play every possible situation, without any knowledge of the distribution of the rest of the cards in the deck. Following is the basic strategy for four of more decks when the dealer stands on soft 17. Links are also provided to more basic strategy charts for other rules. However the most common game is the multiple deck game (4 or more decks) where the dealer stands on soft 17. If you only memorize one table I would suggest the one below. It can be played effectively under any rules.

To use the basic strategy look up your hand along the left vertical edge and the dealer’s up card along the top. In both cases an A stands for ace. From top to bottom are the hard totals, soft totals, and splittable hands. Rule variations can have an effect on some borderline situations. The most flexible rules are the number of decks, whether the dealer hits or stands on a soft 17, and whether doubling is allowed after splitting.

The best way, I have found, to memorize the basic strategy is notice patterns and to try to understand why you should play every situation as the chart says. Then make flash cards and go over and over them until you know it cold. Do not deal out cards to yourself because the soft totals and the pairs will not occur often enough to test your knowledge.

Many people do not believe in the basic strategy because they once took the advice of someone who knew it and then lost the hand. Let me make something perfectly clear, you will not win every hand with the basic strategy! In fact you won’t even win half your hands. However I can personally testify that while you will have short term ups and downs over the long run you will roughly break even using it.

Insurance

Do not take insurance, even if you have a blackjack. Card counters can get away with making smart insurance bets when the deck is rich in tens but the non-counting player should always decline it. The following table shows the house edge on the insurance bet depending on the number of decks used.

Surrender

Sometimes in a 6 or 8 deck game the player has the option to surrender. To be specific the player may forfeit half their bet to be excused from playing out their hand. Obviously this option should only be taken in the worst hands when the net return is less than 50%. In late surrender (the only kind I have ever seen) the player may only opt to surrender after the dealer checks for a blackjack. The following table is a basic guide for when to surrender (Y=yes, N=no).

Appendix 6 shows more details depending on exact card composition and the number of decks.

House Edge

See my Blackjack House Edge Calculator to determine the house edge under 360 possible rule combinations.

Rule Variations

Rule variations will have an effect on the player’s expected return. The numbers below show the effect on the player’s return under various rules and after taking into consideration proper basic strategy adjustments. These changes are relative to the following Atlantic City rules: 8 decks, dealer stands on soft 17, player may double on any first two cards, player can double after splitting, player may resplit to 4 hands.

A “five card Charlie” is an automatic winner on any five card hand that has not busted. This rule does not apply after splitting or if the dealer has a blackjack. A few days in 1999 Binion’s Horseshoe offered this generous rule (I was fortunate to be there at the time).In European blackjack and in many online casinos the dealer does not check for a blackjack after dealing the cards. This is referred to as the “no-peek” rule, because the dealer doesn’t peek to see if he has a blackjack. In this case, after the players have played their hands, if the dealer does have a blackjack, then the player loses the full amount bet, including the additional bet if the player doubled or split.

Beware 6-5 Single Deck Blackjack

Many casinos in Las Vegas have prominent signs saying “Single Deck” blackjack. However on a much smaller sign it says “Blackjack pays 6 to 5.” Aside from the 6-5 rule the house edge would be 0.05%. However the 6-5 on blackjacks costs the player an additional 1.39%, for a total house edge of 1.44%. At some tables a player blackjack always wins, which lowers the house edge by 0.21% to 1.23%. Either way this is by far the worst blackjack game in the city and I urge you to avoid it.

Bad Strategies

Three popular bad strategies encountered at the blackjack table are never bust, mimic the dealer, and always assume the dealer has a ten in the hole. All three of these are very bad strategies. Following are my specific comments on each of them, including the house edge under Atlantic City rules (dealer stands on soft 17, split up to 4 hands, double after split, double any two cards) of 0.43%.

Never bust: For my analysis of this strategy I assumed the player would never hit a hard 12 or more and based all other decisions on maximizing expected value under this assumption. This results in a house edge of 3.91%.

Mimic the dealer: For my analysis of this strategy I assumed the player would always hit 16 or less and stand on 17 or more. The player as well as dealer stood on soft 17. The player never doubled or split, since the dealer is not allowed to do so. This results in a house edge of 5.48%.

Assume ten in the hole: For this strategy I first figured out the optimal basic strategy under this assumption. If the dealer had an ace up I reverted to the proper basic strategy assumption of assuming the dealer did not have a ten. Then I went back and used this strategy under regular playing conditions. This results in a house edge of 10.03%.

Card Counting

Let me say loud and clear that card counting is hard and is not as rewarding as television and the movies make it out to be. If it were an easy way to make money everyone would be doing it.

If you do not know the basic strategy trying to count cards is highly ill-advised. Experienced card counters still play by the basic strategy the great majority of the time. There can be no short cut around learning the basic strategy, those who attempt card counting without a firm foundation in the basic strategy are making a big mistake.

To be a successful counter you have to be able to count down a deck fast and memorize large tables of numbers as well as make it look like you’re just a casual player. Furthermore, with today’s rules, a realistic advantage the counter will have is only 0.5% to 1.5%. You will not win money slowly and gradually but your bankroll will go up and down like a roller coaster in the short run. Only in the long run over hundreds of hours of playing can you count on winning.

The underlying principle behind card counting is that a deck rich in tens and aces is good for the player, a deck rich in small cards is good for the dealer. The reason for this is complicated but to give just two examples: (1) Blackjacks are more common in ten and ace rich decks, which benefit the player more than the dealer. (2) The probability of busting a stiff hand is greater. The dealer is forced to hit a stiff hand and the player is not.

To gauge the richness of the deck in good cards the player will keep track of the cards the are already played. Strategies vary but all assign a point value to each card. For example the Ken Uston’s Plus/Minus strategy assigns a value of +1 to 3, 4, 5, 6 and 7, and -1 to tens and aces. Everything else is 0, or neutral. At the beginning of a deck or shoe the count is 0. Then the counter constantly adds and subtracts from the count according to the cards played. This running total is called the “running count.” A positive count means that a disproportional number of small cards have already been played which means the deck is rich in large cards. To determine the “true count” divide the running count by the number of decks left to be played, or in some strategies the number of half decks. This will tell you the relative richness of the deck in good cards.

The true count is used in two ways, to determine how much to bet and how to play your hand. Unless it is obvious every situation has a line in which you should play one way if the count is above the line and another if below. For example a 12 against a 6 may dictate that you stand if the true count is +1 or greater and hit if the true count is less than +1. The counter will also bet more when the deck is rich in good cards.

A problem arises when it comes to treating aces. The player should bet more when the deck is rich in aces since they add to the probability of getting a blackjack. However, when it comes to playing your hand the number of aces left is not nearly as important as the number of tens, so it is desirable, but not necessary, to distinguish between tens and aces. Some card counting strategies keep a side count of aces. In the Hi-Opt I and Revere Plus/Minus aces are counted separately and only considered when making the wager. This is a more accurate and powerful way to play than assigning a negative value to aces and not keeping a side count, as some strategies do. Yet many people feel that for the beginner it is too confusing to keep two counts. A player is more likely to make mistakes keeping two counts and that costs money. The efficiency of a strategy that does not keep a side count of aces is only modestly less but you likely will gain more from fewer mistakes made. Different experts fall in various places in the spectrum in terms of what to recommend for the beginner. The Zen Count takes the middle ground and gives aces a value of -1 and tens -2. Personally I have tried both and would recommend against a count that requires a side count of aces to a person ready to take up card counting. The Uston Advanced Plus/Minus is a good strategy that does not involve an ace side count and can be found in the book Million Dollar Blackjack. How well you know a counting strategy is much more important that which strategy you know.

Legally speaking the player may play blackjack any way he wants without cheating or using a computer, and the casinos may do anything from making conditions unfavorable to barring in an effort to stop anyone who they deem has an advantage over the game. Much of the challenge of card counting is avoiding suspicion that you are anything but a normal non-counting player. The most obvious indication that somebody is counting is that they make a substantial increase in bet size after a lot of small cards leave the table. Although the greater the factor by which you can increase your bet the greater your odds of winning, more than doubling your last bet is a fast way to arouse “heat” from the dealer and pit boss. Usually when casinos do realize you are counting they will either shuffle the cards whenever you increase your bet, essentially removing any advantage, or ask you to leave.

This is only scraping the surface of the subject of card counting. To learn more see my book review section for suggestions on good blackjack books.

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o Focus on the dealer and his up card, not your neighbor’s cards. If other players at the table are upsetting you with their playing strategies, you can move.

o Drinking and gambling do not mix. Players have been known to lose thousands waiting for their free drink.

o You must have discipline, so set loss limits and win goals.

o Only bet what you can afford to lose. Rent, car payments and other day-to-day expenses have no place in a casino.

o Even the small blackjack player deserves to be rewarded for her play. There is nothing wrong with asking for a free breakfast, lunch or something smaller like a deck of cards or a pair of dice. But since pit personnel are most likely not tracking your play, you will need to ask.

o Most players don’t even consider the rules before they sit down and play. If you don’t know the rules or any of the playing stratagies of blackjack, learn the game on a 25-cent video blackjack machine. A five-dollar blackjack game should not be your classroom.

o When you feel tired, it’s time to call it quits, or at least take a break and rest for an hour or two.

o Compared to a single deck, a two-deck game handicaps your play -0.35%, four decks, -0.48%, six decks, -0.54% and eight decks -0.58%. As you can see, it is always to your advantage to play on a game that offers the fewest decks. Also note, the house edge goes up substantially when you go from one deck to two, but the change is less dramatic as you add more decks. How much is this costing you in dollars and cents? If you were to play 100 hands per hour at $5 per hand, each -0.1% would cost you approximately 50¢ per hour. Playing on a game with two decks versus one deck will cost you $1.75 per hour, with each additional deck costing you increasingly more.

o Smart blackjack players always play in a casino that offers the best rules. To avoid hostile playing conditions in blackjack, look for the following combination of rules that are favorable to the player:

a single deck game
surrender, both early and late
double down allowed on any two cards
double down allowed after splitting pairs
multiple pair splitting allowed, plus resplitting aces
dealer stands on a soft 17
deep deck penetration

Originially posted on BetFirms

o Stay away from all those proposition bets in the center of the layout. The dealer who is barking their performance is doing nothing more than inducing play on wagers that are all “house bets”, carrying a casino advantage as high as 16%.

o Once you learn how the Odds wager works (another column, another day), never make a Line or Come bet that you are not willing to back up with full odds.

o If the table minimum is way over your head, you shouldn’t be on that game. Bet with your head, not over it.

o If you need help, ask a dealer. Most will be willing and able to help you.

o You are responsible to pick up your winnings. As they say, “they lay, they play.”

o A Pass Line bet with two Come bets is plenty of action on this energetic game. Don’t get so caught up in the game and start throwing money at the dealer for his “requested (high house percentage) bets.”

o Do not try to hand cash to the dealer to make change. The dealer is not allowed to take any cash or chips directly from the customer. You need to place your money on the layout, before the shooter gets the dice, and ask the dealer for “change only.”

o Some wagers like Pass/Don’t Pass bets, Odds, Come wagers, the Big 6/8 or Field bets can be made by you. On the other wagers, place you money on the layout and ask the dealer to make those bets for you.

o Keep your hands off the table and out of the way of the dice being thrown. You do not want to disrupt the game by altering the toss with your hands.

o Tables have rails all the way around for storing your gaming chips. Use them. Also, underneath, there is shelving for your drinks.

o If you are the shooter, give the dice a good toss across the table. Never try to slide dice across the layout thinking you can control the outcome. The first time the boxman will call out, “No roll.” The second time, possibly some token punishment like a slapped wrist. The third-adios Sally.

o If you are a Don’t Pass bettor, don’t scream out, “Come on, seven!” Betting against the majority of players is bad enough, but rooting against them and gloating after a win is awful form.

* Watch out for the Sucker Bets: proposition bets like the seven have what seem to be high payoffs, but the house edge is just too high for us to recommend you lay down your wagers on them.  The seven is a sucker bet because it is the worst wager on a crap game.  The house edge is up to 16.7% on this one, which is extremely high.

You should take a look at the pass line wager, which has a measly 1.41% for a house edge.  There is where the value is on the craps table.

Originially posted on BetFirms

One difference between roulette and all other table games is that roulette chips have no value denomination printed on them. This actually is the true definition of a chip, one which indicates its value is technically called a check. The roulette table comes with six to eight sets of different colored chips, each set consisting of 300 chips. When a player buys-in they get their own color and the value would be the buy-in divided by the number of chips received. The dealer will place a token on top of the dealer’s stack of that color of chips to indicate the value.

United States Rules

A U.S. roulette wheel consists of 38 numbered slots: numbers 1 to 36, a zero, and a double zero. The betting layout consists of every individual number as well as a host of “outside” or combinations of numbers. After the players make their bets the dealer spins the wheel and a ball and after several seconds the ball will land in one of the numbered slots.

The following table displays the available bets, the payoff, and the probability of winning under U.S. rules. All casinos in the U.S. follow these rules except for in Atlantic City. The house edge on all bets is 1/19, or 5.26%, except for one bet. The exception is the 0-00-1-2-3 combination, which carries a house edge of 7.89%.

Single Zero Rules

A few places in the U.S. offer single zero roulette. This lowers the edge on every bet is 1/37, or 2.70%. The lists below shows some places that I have seen or heard have single zero roulette. The minimums at the single zero wheels are usually higher than the double zero wheels and are open fewer hours. Often the single zero wheels are found in the high roller rooms.

Las Vegas

• The Tropicana
• The Monte Carlo
• The Stratosphere
• Mandalay Bay
• Venetian
• Aladdin
• Luxor
• Nevada Palace (ten cent minimum)

In addition the Bellagio and Mirage offer single zero roulette with the Hamburg variation of the European half back rule (see below). This means if the player makes an even money bet and the ball lands in zero the player will only lose half. This lowers the house edge to 1.35%. The Bellagio has one such wheel with a $100 minimum. The Mirage has two with minimums of $25 and $100.

Atlantic City

• Hilton
• Bally’s Park Place
• Harrah’s
• Caesars
• Sands
• Tropicana
• Showboat
• Taj Majal
• Trump Marina

All Other

• Grand, Tunica, Mississippi

Atlantic City Rules

In Atlantic City any all even money bets (red, black, odd, even, 1-18, 19-36) follow a variation of the European imprisonment rule (see below). In the event the ball lands in 0 or 00 the player will lose only half of any even money bet. This lowers the house edge to 2.63% on these bets. This rule does not apply on single zero wheels.

European Rules

European roulette is played on a single wheel and also features a favorable “en prison” rule. Under this rule if the player makes any even money bet (red, black, odd, even, 1-18, 19-36) and the ball lands in zero the player either gets half the bet back or it becomes inprisoned. If an imprisoned bet wins on the next spin it is released and the player gets it back, without winnings. What is subject to debate, or rule variation, is what happens to an imprisoned bet if the ball lands in zero on the next spin. I have heard of four different rule variations and each are explained below.

1. At a casino I visited in Hamburg half of even money bets are returned if the ball lands in zero. This house edge is 1.3514%.
2. At a casino I visited in Berlin even money bets became imprisoned if the ball landed in zero. The bet was returned if it won on the next spin. If a zero occured on the second spin the bet lost. The rule card in Berlin said the player may request the croupier to move the imprisoned bet from one even money bet to another. This house edge is 1.3879%.
3. According to the American Mensa Guide to Casino Gambling an imprisoned bet remains imprisoned until a non-zero occurs. This house edge is 1.3514%.
4. According to a dealer from Holland who wrote me if two zeros occurs in a row the bet becomes double imprisoned. In this case two winning bets in a row must occur to release it. If a anything else occurs, including more zeros, the bet is lost. This house edge is 1.3706%. The dealer said that the player also has the option to get half back immediately, which is the better option.

Number Placement

To the casual observer it would appear that the numbers on the wheel are not organized and seem to be distributed randomly. The only obvious patterns are that red and black numbers alternate and that usually two odd numbers alternate with two even numbers. However the distribution of numbers was carefully arranged so that the sum of the numbers for any given section of the wheel would be roughly equal to any other section of equal size. Most numbers are part of a pair, with one number between them. These pairs add to either 37 or 39.

Wheel Tracking

There are books out there that promise big riches by tracking the outcomes on a roulette wheel over thousands of spins and then examining the distribution to see if the wheel is biased toward one side. I don’t deny that this has been done in the past but I doubt it is a practical way to beat the casinos today. There are products the casinos can use today that track every spin and on a periodic basis run the distribution through a statistical test to check for bias. Although I am not an expert on the subject I believe wheel tracking to be a waste of time today.

Betting Systems

Ugh. If you think you can beat roulette with a betting system please read my section debunking betting systems. If you don’t believe what I say there here is what the Encyclopedia Britanica says under the subject of roulette:

The oldest and most common betting system is the Martingale or “doubling-up” system, in which bets are doubled progressively. This probably dates back to the invention of the Roulette wheel, but every day of the week some gambler somewhere reinvents it, or some variation of it, and believes he has something new. Over the years hundreds of “sure-fire” winning systems have been dreamed up, but regardless of what system is used, in the long run it cannot overcome the house’s advantage of the 0, or 0 and 00. This house advantage is the only system that consistently wins in the long run.

Better known as “The Wizard of Odds”, Michael Shackleford uses math and computer analysis to determine optimal playing strategy for all casino games. His work can be found on his website, WizardofOdds.com with player strategies and probabilities on most of the casino games. Michael lives in Las Vegas and in his spare time likes to college license plates and gamble.

You can read more about Michael’s work at his website, WizardofOdds.com

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“A common gamblers’ fallacy called ‘the doctrine of the maturity of the chances’ (or ‘Monte Carlo fallacy’) falsely assumes that each play in a game of chance is not independent of the others and that a series of outcomes of one sort should be balanced in the short run by other possibilities. A number of ‘systems’ have been invented by gamblers based largely on this fallacy; casino operators are happy to encourage the use of such systems and to exploit any gambler’s neglect of the strict rules of probability and independent plays.”

Probability and Measure (second edition) by Patrick Billingsley provides specific proof that betting systems do not win in the long run. On page 94 he states that, “…There are schemes that go beyond selection systems and tell the gambler not only whether to bet but how much. Gamblers frequently contrive or adopt such schemes in the confident expectation that they can, by pure force of arithmetic, counter the most adverse workings of chance…” After proving a detailed explanation regarding the proof, he concludes (page 95), “…Thus no betting system can convert a subfair game into a profitable enterprise…”

The Martingale (double after every loss) system

Every week I receive two or three e-mails asking me about the betting system by which a player doubles his/her bet after a loss. This system is generally played with an even money game such as the red/black bet in roulette or the pass/don’t pass in craps and is known as the Martingale. The reasoning behind those who believe it is that by doubling your bet after a loss, you will always win enough to cover all past losses plus one unit. For example if a player starts at $1 and loses four bets in a row, winning on the fifth, he will have lost $1+$2+$4+$8 = $15 on the four losing bets and won $16 on the fifth bet. The losses were covered and he had a profit of $1. The fallacy is the promise of guaranteed winnings by using the Martingale. Occasionally the player will lose several bets in a row and will reach a point where he doesn’t have enough money to double.In order to prove this point, I created a program that simulated two systems, the Martingale and flat betting, and applied each by betting on the pass line in craps (which has a 49.29% of winning). The Martingale bettor would always start with a $1 bet and start the session with $255 which is enough to cover 8 losses in a row. The flat bettor would bet $1 every time. The Martingale player would play for 100 bets, or until he couldn’t cover the amount of a bet.In that case he would stop playing and leave with the money he had left. In the event his 100th bet was a loss, he would keep betting until he either won a bet or couldn’t cover the next bet. The person flat betting would play 100 bets every time. I repeated this experiment for 1,000,000 sessions for both systems and tabulated the results. The graph below shows the results:

As you can see the flat bettor has a bell curve with a peak at a loss of $1, and never strays very far from that peak. Usually the Martingale bettor would show a profit represented by the bell curve on the far right, peaking at $51, however on the far left we see those times when he couldn’t cover a bet and walked away with a substantial loss. That happened for 19.65% of the sessions. Many believers in the Martingale mistakenly believe that the many wins will more than cover the few loses.In this experiment the average session loss for the flat bettor was $1.12, but was $4.20 for the Martingale bettor. In both cases the ratio of money lost to money won was very close to 7/495, which is the house edge on the pass line bet in craps. This is not coincidental. No matter what system is used in the long run this ratio will always approach the house edge. To prove this point consider the Martingale player on the pass line in craps who only desires to win $1, starts with a bet of $1, and has a bankroll of $2047 to cover as many as 10 consecutive losses. The table below shows all possible outcomes with each probability, expected bet, and return.

The expected bet is the product of the total bet and the probability. Likewise, the expected return is the product of the total return and the probability. The last row shows this Martingale bettor to have had an average total bet of 11.81172639 and an average loss of 0.16703451. Dividing the average loss by the average bet yields .01414141. We now divide 7 by 495 ( the house edge on the pass line) and we again get 0.01414141! This shows that the Martingale is neither better nor worse than flat betting when measured by the ratio of expected loss to expected bet. All betting systems are equal to flat betting when compared this way, as they should be. In other words, all betting systems are equally worthless.

Another experiment

Here is another experiment I conducted earlier which proves the same thing as the experiment above. This one is played against roulette teasting three different systems. Player 1 flat bet a $1 each time. He was not using a betting system. Player 2 started a series of trials with a bet of $1 and increased his wager by $1 after every winning bet. A lost bet would constitute the end of a series and the next bet would be $1. Player 3 also started a series of bets with a bet of $1 but used a doubling strategy in that after a losing bet of $x he would bet $2x (the Martingale). A winning bet would constitute the end of a series and the next bet would be $1. To make it realistic I put a maximum bet on player 3 of $200.Below are the results of that experiment:Player 1

• Total amount wagered = $1,000,000,000
• Average wager = $1.00
• Total loss = $52,667,912
• Expected loss = $52,631,579
• Ratio of loss to money wagered = .052668

Player 2

• Total amount wagered = $1,899,943,349
• Average wager = $1.90
• Total loss = $100,056,549
• Expected loss = $99,997,018
• Ratio of loss to money wagered = .052663

Player 3

• Total amount wagered = $5,744,751,450
• Average wager = $5.74
• Total loss = $302,679,372
• Expected loss = $302,355,340
• Ratio of loss to money wagered = .052688

As you can see the ratio of money lost to money wagered is always close to the normal house advantage of 1/19 =~ .052632 . In conclusion varying of bet size depending on recent past wins or losses makes no difference in outcome in the long run than always betting the same.

A third experiment

An Old Timer’s Guide to Beating the Craps Table is a betting system that makes big promises about turning the craps tables into your own personal cash register. I offered to test his system for free. The results are posted here.

The cancellation betting system

Despite all my warnings about betting systems, readers continually ask me to suggest one. To satisfy those who enjoy playing systems I have done a full explanation and analysis of the cancellation betting system.

Don’t waste your money

The Internet is full of people with betting systems for sale that usually make promises of beating the casino at games of luck. Those who sell these systems are the present day equivalent of the 19th century snake oil salesman. Under no circumstances should you waste one penny on any gambling system. Every time one has been put to a computer simulation it failed and showed the same ratio of losses to money bet as flat betting. If you ask a system salesman about this you likely will get a reply such as, in real life nobody plays millions of trials in the casino. You’re likely to also hear that his/her system works in real life, but not when used against a computer simulation. It is interesting that professionals use computers to model real life problems in just about every field of study, yet when it comes to betting systems computer analysis becomes “…worthless and unreliable…” as the salesman of one system put it.A more basic argument against systems is that if they worked so well why are the salesmen selling them in the first place? If I had a system with just a 1% edge, I could easily turn $1,000 into a $1,000,000 within a year, and keep going. I certainly wouldn’t want any competition to create undo suspicion. If you ask them about this, the answer will usually be that they are trying to transfer money from the evil casinos to give to nice simple folks as yourselves. Gee, how nice of them!

Gambling systems have been around for as long as gambling has. No system has ever been proven to work. From an inside source I know that system salesmen go from selling one kind of system to another. It is a dirty business by which they steal ideas from each other, and are always attempting to rehash old systems as something new.

Here are some examples of system salesmen who try to take advantage of the mathematically challenged. There are hundreds of sites like these on the Internet, and this list is just a sampling. Frequently these sites vanish in the middle of the night, or suddenly direct traffic to a porn site. Please do let me know if any of these links don’t work or take you to other than the intended place.

• Power Craps.
• Ultimate Craps System.
• Gambler’s Bookcase.
• Let’s Talk Winning.

Also be warned that there are many others out there selling get rich quick gambling schemes that claim they are not betting systems. These sites usually throw out lots of fancy physics words like “chaos” and “fractals” but display no evidence they know what these words mean. I have in the past listed some such sites above but got angry letters claiming I shouldn’t criticize what I don’t understand. Personally I feel that every method claiming an easy way to beat the casinos is a scam and I don’t need to understand whatever the secret is. However to be totally fair I’ll only list betting systems above since those have been mathematically debunked by computer simulations. If anyone did find a true easy way to beat the casinos why aren’t they getting rich doing it?

Want some more examples? Visit Shameful Sellers of Crap.

The Wizard of Odds Challenge

Despite all the evidence I have supplied I continue to get challenges via e-mail from people who believe their individual systems are the exception to the rule and really work. These challenges usually come with a request that I waste my time writing a free simulation program to prove or disprove the author’s system. Contrary to what some may think, I do not sit around all day with nothing to do but test betting systems. So in an effort to give the true believer a fair chance to prove me wrong, I will bet your $2,000 against my $20,000 that your system won’t win a negative expectation game over a billion trials. Here are the specific rules:

• The system must be clearly explained. It should not have vague run on sentences, but explain exactly how much to bet at each and every step, and under exactly what conditions. There can be no guesswork or human judgment involved. I must be able to code it into a computer.
• The system must have a betting range of no more than 1 to 1028 units. So a bet of at least 1 unit must be placed on every trial and the maximum bet must be 1028 units.
• The system must be played on craps, roulette, or baccarat under ordinary United States rules.
• The $2000 fee must be paid up front.
• After the fee is paid I will program the system in C++. A random simulation will then be performed of one billion bets resolved. If the player shows a profit at the end you win, if the player shows a loss I win. Should you win, I’ll state publicly, on my main page, that your system defies the laws of probability and that I was wrong.
• The results of my simulation are binding. If you feel I did not run an honest simulation, I will provide the source code and you can have your own experts challenge it.

Please, Don’t Write

The most frequent topics about which people try to debate with me are about betting systems. I have said everything I have to say about them above. Those wasting my time trying to argue that the dice or roulette ball has a memory will get a brush off at best, or, more likely, no reply at all. If you really want to discuss this topic one on one with me, you will have to pay $200 per hour for my time, with a retainer fee up front.

Better known as “The Wizard of Odds”, Michael Shackleford uses math and computer analysis to determine optimal playing strategy for all casino games. His work can be found on his website, WizardofOdds.com with player strategies and probabilities on most of the casino games. Michael lives in Las Vegas and in his spare time likes to college license plates and gamble.

You can read more about Michael’s work at his website, WizardofOdds.com

Originially posted on BetFirms

Basic Strategy

Following is a strategy for the payoff table above. To determine the best play look under the list given the number of deuces hold (you never discard a deuce). Then look for the highest playable hand on the list. For example if you have both a pair and four to a flush you would keep the pair because it is higher on the list. The numbers on the right represent the expected return, which can vary depending on the discards.

The player should evaluate straight flush draws carefully. Usually the number of ranks the straight flush spans determines the spread. However if the straight flush is on the low end this can reduce the number of ways to complete it. For example suited 3-4-5

0 Deuces

1. Natural royal flush (800.0000)
2. 4 to a royal flush (19.574469)
3. Straight flush (9.0000)
4. Four of a kind (5.8510637)
5. Full house (3.0000)
6. Three of a kind (2.0175762)
7. Straight/Flush (2.0000)
8. 4 to an outside straight flush (1.65958)
9. Suited 10-J-Q (1.3987)
10. 4 to an inside straight flush (1.3829787)
11. 3 to a royal flush, except 10-J-Q (1.2719704)
12. Pair (0.560222)
13. Two pair (0.5106383)
14. 4 to a flush (0.5106383)
15. 4 to an outside straight (0.5106383)
16. 3 to a straight flush, spread 3-4, suited 345 (0.5051-0.3959)
17. 2 to a royal flush, jack highest¹ (0.38815913)
18. 3 to a straight flush, spread 5 and suited 346,356 ^2,3 (0.35522664)
19. 4 to an inside straight, except missing deuce (0.34042552)
20. 2 to a royal flush, queen highest⁴ (0.33851373)
21. 2 to a royal flush, king highest, no penalty cards⁵ (0.3278446)
22. Garbage, everything discarded (0.32552597)
23. 2 to a royal flush, king highest, 1 penalty card⁶ (0.3185939)
24. 2 to a royal flush, ace highest (0.29768732)
25. 3 to a straight flush, A low (0.2729)

Notes:
1: Suited 7-10-J beats suited 10-J if either (1) king and ace discards, or (2) queen discard without 8

2: Play 4 to an inside straight over 3 to a straight flush, spread 5, if there is a straight penalty card to the straight flush.

3: Play 2 to a royal, queen high, over 3 to a straight flush, spread 5, when 6 or 7 high.

4: Play 2 to a royal, queen high, over 4 to an inside straight if there are no flush penalty cards and a fully open straight possibility. A fully open straight possibility means that there exists some possible straight with no penalty cards discarded. For example with suited 10-Q and unsuited 5-K-A the player has a full chance at a Q-J-10-9-8 straight. However with suited 10-Q and unsuited K-9-5 the player should go for the inside straight because the K and 9 block any fully open straight possibility.

5: There are some rare exceptions in which it is better to discard everything rather than keep two to a royal, king high, with no penalty cards. See my deuces wild appendix 1 for a list of these exceptions.

6. There are some rare exceptions in which it is better to keep the two to a royal, king high, despite a 9 or ace straight penalty card. See my deuces wild appendix 3 for a list of these exceptions.

1 Deuce

1. Wild royal flush (25.0000)
2. 5 of a kind (15.0000)
3. Straight flush (9.0000)
4. Four of a kind (5.8510637)
5. 4 to a royal flush (3.4042554)
6. Full house (3.0000)
7. 3 consecutive suited cards 5-6-7 or greater (2.21277)
8. 3 of a kind (2.01758)
9. Straight (2.0000)
10. Flush (2.0000)
11. All other 4 to a straight flush (1.70213 to 1.97872)
12. 3 to a royal flush, highest card king or less (1.1424607)
13. 2 consecutive suited cards, 6-7 or higher, + deuce (1.0952822)
14. 3 to a royal flush, ace highest card, no penalty cards (1.0462534)
15. Deuce only (1.0328652)
16. 3 to a royal flush, ace highest card, 1+ penalty card³ (1.0286771)
17. 2 consecutive suited cards, 5-6 or lower, + deuce (1.0166513)
18. 4 to an outside straight (1.0000)

Note:
3: There are some rare exceptions to this rule that cut both ways. Sometimes with two a royal with no penalty cards it is better to hold the deuce only, and sometimes with a penalty card you should go for the royal. See my deuces wild appendix 2 for a list of these exceptions.

2 Deuces

1. Wild royal flush (25.0000)
2. 5 of a kind (15.0000)
3. Straight flush (9.0000)
4. Four of a kind (5.8510637)
5. 4 to a royal flush (4.617021)
6. 2 consecutive suited cards, 6-7 or higher, + deuces (3.3404255)
7. 2 deuces only (3.2730188)
8. 2 consecutive suited cards, 5-6 of lower, + deuces (3.1276596)
9. Full house or less

3 Deuces

1. Wild royal flush (25.0000)
2. 3 deuces only, non-pair discarded (15.059204)
3. 3 deuces only, pair 9 or less discarded (15.057354)
4. 5 of a kind (15.0000)
5. 3 deuces only, pair 10 or greater discarded (14.938946)
6. 4 to a royal flush (11.829787)
7. Straight flush or less

4 Deuces

1. 4 deuces (200.0000)

Terms:

Outside straight: An open ended straight that can be completed at either end, such as (7,8,9,10). A wild card shall not be used to complete an outside straight. For example 2,5,7,8 is not 4 to an outside straight because of the missing 6. However 2,5,6,7 is 4 to an outside straight.

Inside straight: A straight with a missing inside card, such as (6,7,9,10).

Penalty card: A penalty card is a potentially useful discarded card. For example if the player had 3 to a royal and 4 to a flush the correct play is to keep three to the royal, discarding the fourth suited card. The discarded suited card would be called a flush penalty card because it could have been used to complete a flush. By discarding it the players odds of forming a flush are “penalized.” Sometimes penalty cards can affect borderline plays. For example if the player had a suited 10 and king, with no other cards of that suit, nor a 9, jack, queen, or ace, then the player should keep the two to a royal flush. However this is only marginally better than discarding everything. If the player had just one suited card, or any card that could be used to complete a straight, then the odds of forming a flush or straight would be depressed, lowering the overall expected return below that of discarding everything. So in that situation the player should be mindful of the effect of penalty cards.

Methodology

To determine the above strategy I created a program can determine the expected return of the best play of any hand. The way it works is to consider all 32 ways to play a hand. For every play the program systematically scores the held cards with every possible set of discards and averages the results. The play that yields the greatest average is determined to be the best play and the specific statistics for that play are displayed. The program can also show the statistics for non-optimal plays. Using this program it was then a time consuming task to try numerous borderline hands and rank them in order of expected return. I used Bob Dancer’s Deuces Wild Video Poker report to verify my strategy. There I found some obscure exceptions that I did not notice, which I used to correct my strategy. So I would like to thank Bob Dancer for his help. You may order his software and strategy cards here.

Better known as “The Wizard of Odds”, Michael Shackleford uses math and computer analysis to determine optimal playing strategy for all casino games. His work can be found on his website, WizardofOdds.com with player strategies and probabilities on most of the casino games. Michael lives in Las Vegas and in his spare time likes to college license plates and gamble.

You can read more about Michael’s work at his website, WizardofOdds.com

Originially posted on BetFirms

0 deuces

1. 4,5 to a royal flush
2. Any pat paying hand
3. 4 to a straight flush
4. 3 to a royal flush
5. Pair
6. 4 to a flush
7. 4 to an outside straight
8. 3 to a straight flush
9. 4 to an inside straight, except missing deuce
10. 2 to a royal flush, J,Q high

1 deuce

1. Any pat four of a kind or higher
2. 4 to a royal flush
3. Full house
4. 4 to a straight flush with 3 consecutive singletons, 5-7 or higher
5. 3 of a kind, straight, or flush
6. All other 4 to a straight flush
7. 3 to a royal flush
8. 3 to a straight flush with 2 consecutive singletons, 6-7 or higher
9. deuce only

2 deuces

1. Any pat four of a kind or higher
2. 4 to a royal flush
3. 4 to a straight flush with 2 consecutive singletons, 6-7 or higher
4. 2 deuces only

3 deuces

1. Pat royal flush
2. 3 deuces only

4 deuces

1. 4 deuces

Terms:

Outside straight: An open ended straight that can be completed at either end, such as the cards 7,8,9,10.

Inside straight: A straight with a missing inside card, such as the cards 6,7,9,10. In addition A,2,3,4 and J,Q,K,A also count as inside straights because they are at an extreme end.

Any hand that does not appear on the above lists should never be played. Most notably a two pair, keeping a single pair has a higher expected value.

Example: Suppose you have the following hand.

The top two plays are (1) keep the pat flush, and (2) keep the 4 to a straight flush. The 4 to a straight flush consists of 3 consecutive singletons with a range of 5-7 or greater. This appears higher on the list than a pat flush, and is thus the better play.

Comparison to Optimal Strategy

The following table compares the probability and return of each hand under both the simple strategy and the optimal strategy.

The next table is a frequency distribution of the error, or difference in expected return, between the simple strategy and the optimal strategy.

Better known as “The Wizard of Odds”, Michael Shackleford uses math and computer analysis to determine optimal playing strategy for all casino games. His work can be found on his website, WizardofOdds.com with player strategies and probabilities on most of the casino games. Michael lives in Las Vegas and in his spare time likes to college license plates and gamble.

You can read more about Michael’s work at his website, WizardofOdds.com

Originially posted on BetFirms