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However there are other ways you get four aces in the same hand, for example the last card might be an 8 or 9. I would have to do a computer simulation to consider all the other combinations.

After performing my own infinite deck analysis for Blackjack with the same rules as yours dealer stands all 17s, re-splitting allowed to 4 hands except Aces, which can only be split once, doubling after splitting, draw only one card to split Aces , I came across your site.

In comparing expected values, I obtained the same numbers as you in all cases, except for pair splitting, which were slightly different. It took me years to get the splitting pairs correct myself.

Cindy of Gambling Tools was very helpful. Resplitting up to four hands is allowed. Here is how I did it. Determine the probability that the player will not get a third eight on either hand.

Go through all ranks, except 8, subtract that card from the deck, play out a hand with that card and an 8, determine the expected value, and multiply by 2.

For each rank determine the probability of that rank, given that the probability of another 8 is zero. Take the dot product of the probability and expected value over each rank.

Multiply this dot product by the probability from step 2. Determine the probability that the player will resplit to 3 hands.

Take another 8 out of the deck. Repeat step 3 but multiply by 3 instead of 2. Multiply dot product from step 7 by probability in step 5.

Determine the probability that the player will resplit to 4 hands. Repeat step 3 but multiply by 4 instead of 2, and this time consider getting an 8 as a third card, corresponding to the situation where the player is forced to stop resplitting.

Multiply dot product from step 11 by probability in step 9. Add values from steps 4, 8, and The hardest part of all this is step 3.

I have a very ugly subroutine full of long formulas I determine using probability trees. It gets especially ugly when the dealer has a 10 or ace up.

Dear wiz, How do you calculate the probability of getting three sevens, three colored sevens, and three suited sevens in blackjack? The number of ways to draw 3 suited sevens is the number of suits 4 times the number of ways to choose 3 out of 6 sevens of that suit in the shoe.

Good job and well done. Or is this a stupid question, i. Yes, I calculate blackjack odds using a combinatorial approach, analyzing every possible ways the player and dealer cards can come out, taking the greatest expected value at every decision point.

This is harder to program than a simulation but I feel is more elegant and a nice challenge in recursive programming. However I still respect my peers to do simulations.

I recently went to Vegas and had an incredible hand of blackjack Then was dealt blackjack on all 4 hands! What are the odds on this?

It was a 6 card deck shoe, I was sitting in 3 seat of a 4 person game. Assume a fresh shuffle? Not too many places allow resplitting aces, so be glad you were playing somewhere that did.

Your seat position does not matter. I just witnessed a friend get four blackjacks in a row starting with the first hand of a newly shuffled single deck playing head to head against the dealer.

Instead of a decimal probability, could you tell me the odds of this? It must be astronomical. Hope to hear from you.

I seem to get a variation of this question at least once a month. If the probability of something happening is p then the probability of it happening n times in a row is p n.

However the actual probability is much less, because as the player gets each blackjack the ratio of aces to cards left in the deck decreases.

First I wanted to tell you how much I look at and love your web site, and admire your math skills. Thank you very much. Michael, a person asked you if they are not counting cards in blackjack, what difference does it make how many decks are being used.

You stated the difference had mostly to do with the number of stiff hands possible, due to the fact that if a small card came out it was more likely a large card would follow and vice-a-versa.

How could that be? Would it still not be a random event with the possibility of a small or large card coming out being equal, if you are not counting?

Every legitimate blackjack expert agrees the house edge decreases as the number of decks goes down, all other rules being equal.

However it is hard to explain why. First, it is true that you are more likely to get one small card and one big card in single-deck than multiple-deck.

Although stiffs can cut both ways the player has the free will to stand, the dealer must always hit them. At a single deck game what is the probability all three players and the dealer get a blackjack the first round after a shuffle?

Following are the probabilities: There is a lot of useful and interesting info. Where would you suggest that a person interested in writing something similar to your "blackjack house edge calculator" go for more info?

Thank you for your response. Thanks for the compliment. It took me years to get my blackjack engine to work perfectly splits when the dealer had a 10 or ace showing was very tricky.

An easier way to get the house edge for blackjack is to write a random simulation. I am a blackjack dealer and last night I amazed my table on a single-deck blackjack game the horrible 6 to 5.

My hand consisted of an Ace up, Ace in the hole and then I drew the other 2 Aces and then a 7 for 21!

What are the odds of this happening and I am especially interested in knowing the math. In blackjack, what is the probability of the dealer making a stopping hand drawing eight cards?

This happened to a friend of mine online and I think it's an extremely rare occurrence. How about seven cards? Thanks for the great site and keep up the awesome work!

Assuming a six-deck game, where the dealer stands on soft 17, and the player plays basic strategy here are the rounded results based on a million hand simulation.

So the larger the bankroll the better your chances. The house edge will lower the probability of success by an amount that is hard to quantify. For a low house edge game like blackjack, the reduction in the probability of success will be small.

It would take a random simulation to know for sure. Forgive me if I don't bother with that. VegasClick did a small simulation about the probability of success with the Martingale.

As I read your analysis of the Royal Match side bet in blackjack , am I correct that your odds are for the first hand of the shoe?

It seems to me that if the suits get unbalanced in any direction it would slightly lessen the house edge, and the suits will certainly fluctuate through the shoe.

This is not true. The remaining deck needs to be exhibit more than a certain degree of skewness for the odds to swing to the player's favor.

Consider a hypothetical side that pays 3 to 1 for any suited pair in a one-deck game. What all this shows is that if cards are removed at a uniform distribution the odds of winning go down, however at a very skewed distribution the odds go up.

As the deck is played down sometimes your odds get better, and sometimes worse, but in the long run they average out and stay at a I have been a dealer for 27 years and have seen a lot.

One of my favorites was a guy who never looked at his cards playing blackjack I thought he was nuts of course but some days he won and some days he lost.

Just like most people. I tried this myself on a free gambling website and won 2 out of 3 times gambling 20 minute sessions. My question is this: How much worse off are you doing this than trying to play basic strategy?

Under typical Vegas rules 6-deck, dealer hits soft 17 the house edge by always standing is I lost a lot of money playing Cryptologic Blackjack today.

Within 35 hands, the dealer showed a 6 seven times and won each time. This was verified through the logs. At Cryptologic they use 8 decks and the dealer stands on a soft According to my blackjack appendix 2 , the probability of the dealer busting with a 6 up is 0.

So the probability of not busing is 1 - 0. The probability of not busing 7 times out of 7 is 0. First off, my apologies if you consider this a basic math question.

We use six decks. Neither my player or I had ever seen this before. What are the odds of this? I am a pit supervisor at a local casino and recently had a dealer deal two players two seven of clubs each and give himeself the last seven of clubs as his upcard on a five-deck shoe.

What are the odds of five of the same card coming out of a five-deck shoe in order? Assuming liberal Vegas Strip rules six decks, dealer stands on soft 17, double after split allowed, late surrender allowed, resplitting aces allowed the following are the probabilities of each possible outcome when doubling on the initial two cards.

This does not include doubling after splitting. Have you ever heard of such a horror streak? My calculations estimate the probability of 19 straight losses as 1 chance in about ,; you may well correct me on this.

Had I done anything differently, I would have been cleaned out well before the 19 hands came up. From my blackjack appendix 4 we see the following probabilities for each initial hand.

So the probability of going exactly 19 losses in a row is 0. By way of comparison, the probability of being dealt a royal flush in video poker is 1 in ,, or 2.

If in an 8-deck or continuous shuffle blackjack game there is no difference in the probabilities of a card appearing at any time, why have you posted Blackjack Appendix 18?

If the probabilities say hit on 16 vs. I see the change if the deck is shrinking or in a game like Spanish 21 where there is a bonus for 21 with 5 or more cards, but why in an 8-deck game or continuous shuffle?

The reason the strategy changes, according to the number of cards in your hand, as shown in appendix 18, is that every card that leaves the deck changes the probabilities of every card left to be played.

A good example is the single-deck basic strategy says to surrender 7,7 against a 10; but for any other 14 you should hit. Composition-dependent expected returns for 1 to 8 decks Continuous shuffling machine vs.

Value and strategy for and bonuses Risk of ruin statistics Probabilities in the first four cards Value of each initial player card House edge using total-dependent vs.

Basic strategy when dealer exposes both cards The Ace-Five card counting method Basic strategy exceptions for three to six cards Blackjack splitting strategy when a back-player is betting Blackjack doubling strategy when doubling after splitting aces is allowed On This Page Internal Links.

Many casinos also offer up live dealer versions of their games, which end up offering you a land casino experience from the comfort of your own home!

Table game bonuses are pretty rare, as many casinos are attempting to make sure that advantage players do not cheat the casino. With that being said, there are occasional bonuses you will find, and if you play your cards correctly pun intended , you will have a great chance of winning some money.

Nearly all casinos offer signup bonuses, though the clearance rate changes from game to game. For example playthrough on blackjack will likely be a lot higher than that on slots.

Comparing table games to slots is a difficult task, as they are completely different styles of games. We would say that there are some definite advantages to playing table games.

The first of these is that the House edge is much lower for table games compared to video slots odds , meaning that if you stick to basic strategies, you can oftentimes walk away further ahead than if you were playing slots.

Online table games also offer up a bunch of variations that bring a different wrinkle to your gameplay. With slots you can generally only hit the spin button and hope for the best.

Yes, online table games are very safe. Online casinos utilize the best technologies to ensure that user data is safe and the games are fair.

Players should look at and understand the technology they are using to play games with, and rest assured that all of the data they put forth is safe.

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Basic strategy when dealer exposes both cards The Ace-Five card counting method Basic strategy exceptions for three to six cards Blackjack splitting strategy when a back-player is betting Blackjack doubling strategy when doubling after splitting aces is allowed On This Page Internal Links.

Other basic strategy rules: Never take insurance or "even money. If there is no row for splitting fives and tens , then look up your hand as a hard total 10 or Later references to this game are found in France and Spain.

When twenty-one was introduced in the United States, gambling houses offered bonus payouts to stimulate players' interest.

One such bonus was a ten-to-one payout if the player's hand consisted of the ace of spades and a black jack either the jack of clubs or the jack of spades.

This hand was called a "blackjack", and the name stuck to the game even though the ten-to-one bonus was soon withdrawn. In the modern game, a blackjack refers to any hand of an ace plus a ten or face card regardless of suits or colors.

The first scientific and mathematically sound attempt to devise an optimal blackjack playing strategy was revealed in September This paper would become the foundation of all future sound efforts to beat the game of blackjack.

At a casino blackjack table, the dealer faces five to seven playing positions from behind a semicircular table.

Between one and eight standard card decks are shuffled together. At the beginning of each round, up to three players can place their bets in the "betting box" at each position in play.

That is, there could be up to three players at each position at a table in jurisdictions that allow back betting. The player whose bet is at the front of the betting box is deemed to have control over the position, and the dealer will consult the controlling player for playing decisions regarding the hand; the other players of that box are said to "play behind".

Any player is usually allowed to control or bet in as many boxes as desired at a single table, but it is prohibited for an individual to play on more than one table at a time or to place multiple bets within a single box.

Each box is dealt an initial hand of two cards visible to the people playing on it, and often to any other players. The dealer's hand receives its first card face up, and in "hole card" games immediately receives its second card face down the hole card , which the dealer peeks at but does not reveal unless it makes the dealer's hand a blackjack.

Hole card games are sometimes played on tables with a small mirror or electronic sensor that is used to peek securely at the hole card.

In European casinos, "no hole card" games are prevalent; the dealer's second card is neither drawn nor consulted until the players have all played their hands.

Cards are dealt either from one or two handheld decks, from a dealer's shoe , or from a shuffling machine. Single cards are dealt to each wagered-on position clockwise from the dealer's left, followed by a single card to the dealer, followed by an additional card to each of the positions in play.

The players' initial cards may be dealt face up or face down more common in single-deck games. On their turn, players must choose whether to "hit" take a card , "stand" end their turn , "double" double wager, take a single card and finish , "split" if the two cards have the same value, separate them to make two hands or "surrender" give up a half-bet and retire from the game.

Number cards count as their natural value; the jack, queen, and king also known as "face cards" or "pictures" count as 10; aces are valued as either 1 or 11 according to the player's choice.

If the hand value exceeds 21 points, it busts, and all bets on it are immediately forfeit. After all boxes have finished playing, the dealer's hand is resolved by drawing cards until the hand busts or achieves a value of 17 or higher a dealer total of 17 including an ace, or "soft 17", must be drawn to in some games and must stand in others.

The dealer never doubles, splits, or surrenders. If the dealer busts, all remaining player hands win. If the dealer does not bust, each remaining bet wins if its hand is higher than the dealer's, and loses if it is lower.

If a player receives 21 on the 1st and 2nd card it is considered a "natural" or "blackjack" and the player is paid out immediately unless dealer also has a natural, in which case the hand ties.

In the case of a tied score, known as "push" or "standoff", bets are normally returned without adjustment; however, a blackjack beats any hand that is not a blackjack, even one with a value of Wins are paid out at 1: Many casinos today pay blackjacks at less than 3: Blackjack games almost always provide a side bet called insurance, which may be played when dealer's upcard is an ace.

Additional side bets, such as "Dealer Match" which pays when the player's cards match the dealer's up card, are sometimes available.

After receiving an initial two cards, the player has up to four standard options: Each option has a corresponding hand signal. Some games give the player a fifth option, "surrender".

Hand signals are used to assist the " eye in the sky ", a person or video camera located above the table and sometimes concealed behind one-way glass.

The eye in the sky usually makes a video recording of the table, which helps in resolving disputes and identifying dealer mistakes, and is also used to protect the casino against dealers who steal chips or players who cheat.

The recording can further be used to identify advantage players whose activities, while legal, make them undesirable customers.

In the event of a disagreement between a player's hand signals and their words, the hand signal takes precedence.

Each hand may normally "hit" as many times as desired so long as the total is not above hard On reaching 21 including soft 21 , the hand is normally required to stand; busting is an irrevocable loss and the players' wagers are immediately forfeited to the house.

After a bust or a stand, play proceeds to the next hand clockwise around the table. When the last hand has finished being played, the dealer reveals the hole card, and stands or draws further cards according to the rules of the game for dealer drawing.

When the outcome of the dealer's hand is established, any hands with bets remaining on the table are resolved usually in counterclockwise order: If the dealer's upcard is an ace, the player is offered the option of taking "insurance" before the dealer checks the hole card.

Insurance is a side bet that the dealer has blackjack and is treated independently of the main wager. The idea is that the dealer's second card has a fairly high probability nearly one-third to be ten-valued, giving the dealer blackjack and disappointment for the player.

It is attractive although not necessarily wise for the player to insure against the possibility of a dealer blackjack by making a maximum "insurance" bet, in which case the "insurance proceeds" will make up for the concomitant loss on the original bet.

The player may add up to half the value of their original bet to the insurance and these extra chips are placed on a portion of the table usually marked "Insurance pays 2 to 1".

Players with a blackjack may also take insurance, and in taking maximum insurance they commit themselves to winning an amount exactly equal to their main wager, regardless of the dealer's outcome.

Fully insuring a blackjack against blackjack is thus referred to as "taking even money", and paid out immediately, before the dealer's hand is resolved; the players do not need to place more chips for the insurance wager.

Insurance bets are expected to lose money in the long run, because the dealer is likely to have blackjack less than one-third of the time.

However the insurance outcome is strongly anti-correlated with that of the main wager, and if the player's priority is to reduce variation , they might choose to pay for this.

Furthermore, the insurance bet is susceptible to advantage play. It is advantageous to make an insurance bet whenever the hole card has more than a chance of one in three of being a ten.

Advantage play techniques can sometimes identify such situations. In a multi-hand, face-up, single deck game, it is possible to establish whether insurance is a good bet simply by observing the other cards on the table after the deal; even if there are just 2 player hands exposed, and neither of their two initial cards is a ten, then 16 in 47 of the remaining cards are tens, which is larger than 1 in 3, so insurance is a good bet.

This is an elementary example of the family of advantage play techniques known as card counting. Bets to insure against blackjack are slightly less likely to be advantageous than insurance bets in general, since the ten in the player's blackjack makes it less likely that the dealer has blackjack too.

The rules of casino blackjack are generally determined by law or regulation, which establishes certain rule variations allowed at the discretion of the casino.

The rules of any particular game are generally posted on or near the table, failing which there is an expectation that casino staff will provide them on request.

Over variations of blackjack have been documented. As with all casino games, blackjack incorporates a "house edge", a statistical advantage for the casino that is built into the game.

The advantage of the dealer's position in blackjack relative to the player comes from the fact that if the player busts, the player loses, regardless of whether the dealer subsequently busts.

The loss rate of players who deviate from basic strategy through ignorance is generally expected to be greater. Surrender, for those games that allow it, is usually not permitted against a dealer blackjack; if the dealer's first card is an ace or ten, the hole card is checked to make sure there is no blackjack before surrender is offered.

This rule protocol is consequently known as "late" surrender. The alternative, "early" surrender, gives player the option to surrender before the dealer checks for blackjack, or in a no-hole-card game.

Early surrender is much more favorable to the player than late surrender. Most medium-strength hands should be surrendered against a dealer Ace if the hole card has not been checked.

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