The best possible Blackjack hand is an opening deal of an ace with any In many places the dealer's first card is initially dealt face down.

Enjoy!

Probability Theory Basics and Applications - Mathematics of Blackjack. Let us look at the probabilities for a favorable initial hand (the first two cards dealt) to be achieved. The total number of possible combinations for each of the two cards is C(52, 2) = , Any blackjack strategy is based on counting the cards played.

Enjoy!

Software - MORE

Blackjack, formerly also Black Jack and Vingt-Un, is the American member of a global family of If both dealer and player receive a blackjack or any other hands with the same sum, called a "push", no one wins. Card counting is most rewarding near the end of a complete shoe when as few as possible cards remain.

Enjoy!

Blackjack, formerly also Black Jack and Vingt-Un, is the American member of a global family of If both dealer and player receive a blackjack or any other hands with the same sum, called a "push", no one wins. Card counting is most rewarding near the end of a complete shoe when as few as possible cards remain.

Enjoy!

Software - MORE

If you want to win at Blackjack, you need to gain a good understanding of what Many people use “probability” and “odds” as two interchangeable terms but in 2 because there are two possible permutations of cards in a hand of blackjack.

Enjoy!

Software - MORE

If you want to win at Blackjack, you need to gain a good understanding of what Many people use “probability” and “odds” as two interchangeable terms but in 2 because there are two possible permutations of cards in a hand of blackjack.

Enjoy!

Software - MORE

If you want to win at Blackjack, you need to gain a good understanding of what Many people use “probability” and “odds” as two interchangeable terms but in 2 because there are two possible permutations of cards in a hand of blackjack.

Enjoy!

Software - MORE

Blackjack, formerly also Black Jack and Vingt-Un, is the American member of a global family of If both dealer and player receive a blackjack or any other hands with the same sum, called a "push", no one wins. Card counting is most rewarding near the end of a complete shoe when as few as possible cards remain.

Enjoy!

What is the probability of losing the next six hands at blackjack, using basic strategy? Any series of negative expectation bets is guaranteed to create a possible way to beat the odds of any game through a betting strategy.

Enjoy!

The best possible Blackjack hand is an opening deal of an ace with any In many places the dealer's first card is initially dealt face down.

Enjoy!

From my blackjack appendix 7 we see that each 9 removed from a single deck game increases the house edge by 0. If you were to add a card as the dealer you should add a 5, which increases the house edge by 0. So the probability of winning six in a row is 0. Is it that when I sit down at the table, 1 out of my next playing sessions I can expect to have an 8 hand losing streak? This is not even a marginal play. What is important is that you play your cards right. The fewer the decks and the greater the number of cards the more this is true. Cindy of Gambling Tools was very helpful. The following table displays the results. When I said the probability of losing 8 hands in a row is 1 in I meant that starting with the next hand the probability of losing 8 in a row is 1 in The chances of 8 losses in a row over a session are greater the longer the session. My question though is what does that really mean? 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. These expected values consider all the numerous ways the hand can play out. Here is the exact answer for various numbers of decks. Following this rule will result in an extra unit once every hands. So, the best card for the player is the ace and the best for the dealer is the 5. I have a very ugly subroutine full of long formulas I determine using probability trees.{/INSERTKEYS}{/PARAGRAPH} According to my blackjack appendix 4 , the probability of an overall win in blackjack is I'm going to assume you wish to ignore ties for purposes of the streak. Resplitting up to four hands is allowed. All of this assumes flat betting, otherwise the math really gets messy. Multiply dot product from step 11 by probability in step 9. If there were a shuffle between hands the probability would increase substantially. However if you were going to cheat it would be much better to remove an ace, which increases the house edge by 0. If the probability of a blackjack is p then the probability of not getting any blackjacks in 10 hands is 1- 1-p For example in a six deck game the answer would be 1- 0. There are 24 sevens in the shoe. Probability of Blackjack Decks Probability 1 4. Let n be the number of decks. Since this question was submitted, a player held the dice for rolls on May 23, in Atlantic City. Steve from Phoenix, AZ. It is more a matter of degree, the more you play the more your results will approach the house edge. You ask a good question for which there is no firm answer. It depends on the number of decks. Your question however could be rephrased as, "what is the value of the ace, given that the other card is not a ten. I recently replaced my blackjack appendix 4 with some information about the standard deviation which may help. Or does it mean that on any given loss it is a 1 in chance that it was the first of 8 losses coming my way? 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. Take another 8 out of the deck. So standing is the marginally better play. There is no sound bite answer to explain why you should hit. Determine the probability that the player will not get a third eight on either hand. If you want to deviate from the basic strategy here are some borderline plays: 12 against 3, 12 against 4, 13 against 2, 16 against Deviating on these hands will cost you much less. For how to solve the problem yourself, see my MathProblems. That column seemed to put the mathematics to that "feeling" a player can get. If I'm playing for fun then I leave the table when I'm not having fun any longer. As I always say all betting systems are equally worthless so flying by the seat of your pants is just as good as flat betting over the long term. It depends whether there is a shuffle between the blackjacks. Because the sum of a large number of random variables always will approach a bell curve we can use the central limit theorem to get at the answer. I have no problem with increasing your bet when you get a lucky feeling. Besides every once in awhile throwing down a bigger bet just adds to the excitement and for some reason it seems logical that if you have lost a string of hands you are "due" for a win. I know, I know, its some sort of divine intervention betting system I am talking about and no betting system affects the house edge. The standard deviation of one hand is 1. Determine the probability that the player will resplit to 4 hands. Add values from steps 4, 8, and The hardest part of all this is step 3. It would take about 5 years playing blackjack 40 hours a week before this piece of advice saved the player one unit. For the non-card counter it may be assumed that the odds are the same in each new round. I would have to do a computer simulation to consider all the other combinations. It may also be the result of progressive betting or mistakes in strategy. According to my blackjack appendix 9H the expected return of standing is So my hitting you will save 6. For each rank determine the probability of that rank, given that the probability of another 8 is zero. Any basic statistics book should have a standard normal table which will give the Z statistic of 0. In general the variation in the mean is inversely proportional to the square root of the number of hands you play. Streaks, such as the dealer drawing a 5 to a 16, are inevitable but not predictable. Thanks for the kind words. Multiply dot product from step 7 by probability in step 5. The best play for a billion hands is the best play for one hand. From my section on the house edge we find the standard deviation in blackjack to be 1. Here is how I did it. Blackjack is not entirely a game of independent trials like roulette, but the deck is not predisposed to run in streaks. {PARAGRAPH}{INSERTKEYS}This is a typical question one might encounter in an introductory statistics class. You are forgetting that there are two possible orders, either the ace or the ten can be first. Expected Values for 3-card 16 Vs. However there are other ways you get four aces in the same hand, for example the last card might be an 8 or 9. Unless you are counting cards you have the free will to bet as much as you want. Multiply this dot product by the probability from step 2. To test the most likely case to favor hitting, 8 decks and only 3 cards, I ran every possible situation through my combinatorial program. In that case, the probability of a win, given a resolved bet, is The probability of winning n hands is a row is 0. What you have experienced is likely the result of some very bad losing streaks. I hope this answers your question. Take the dot product of the probability and expected value over each rank. Thanks for your kind words. Putting aside some minor effects of deck composition, the dealer who pulled a 5 to a 16 the last five times in a row would be just as likely to do it the next time as the dealer who had been busting on 16 for several hours. Repeat step 3 but multiply by 3 instead of 2. Determine the probability that the player will resplit to 3 hands. When the dealer stands on a soft 17, the dealer will bust about When the dealer hits on a soft 17, the dealer will bust about According to my blackjack appendix 4 , the probability of a net win is However, if we skip ties, the probability is So, the probability of a four wins in a row is 0. It took me years to get the splitting pairs correct myself. The probability of this is 1 in 5,,, For the probability for any number of throws from 1 to , please see my craps survival tables. There are cards remaining in the two decks and 32 are tens.