Blackjack is one of the most advantageous casino games as in it, an efficient However, knowing the odds of winning and the probability of getting a certain card in your hand, there is no need to do the math as your odds of busting are 0%.

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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.

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If the player's first two cards total 21, this is a blackjack and she wins times her Let's call p the total probability of winning a pass line bet (so p is the number.

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The article explains interesting mathematical & probability concepts for Blackjack which can be applied in alliknow.runed in simple english.

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alliknow.ru › science › article › pii.

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Whether the game is in your favor is independent of the betting system. No system of betting can rescue a losing game. You are correct that with Martingale you.

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want to win at Blackjack, you need to gain a good understanding of what the odds this is a purely mathematical game that is all about odds and probabilities.

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The article explains interesting mathematical & probability concepts for Blackjack which can be applied in alliknow.runed in simple english.

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alliknow.ru › science › article › pii.

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The article explains interesting mathematical & probability concepts for Blackjack which can be applied in alliknow.runed in simple english.

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Take another 8 out of the deck. My question though is what does that really mean? So the probability of winning six in a row is 0. Take the dot product of the probability and expected value over each rank. Here is the exact answer for various numbers of decks. There are 24 sevens in the shoe. If there were a shuffle between hands the probability would increase substantially. From my section on the house edge we find the standard deviation in blackjack to be 1. It took me years to get the splitting pairs correct myself. So standing is the marginally better play. Thanks for your kind words. Cindy of Gambling Tools was very helpful. 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? Multiply dot product from step 11 by probability in step 9. All of this assumes flat betting, otherwise the math really gets messy. For how to solve the problem yourself, see my MathProblems. There is no sound bite answer to explain why you should hit. Multiply dot product from step 7 by probability in step 5. Unless you are counting cards you have the free will to bet as much as you want. I hope this answers your question. Expected Values for 3-card 16 Vs. Thanks for the kind words. Multiply this dot product by the probability from step 2. For each rank determine the probability of that rank, given that the probability of another 8 is zero. 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. The standard deviation of one hand is 1. It depends on the number of decks. That column seemed to put the mathematics to that "feeling" a player can get. Here is how I did it. 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? It depends whether there is a shuffle between the blackjacks. Determine the probability that the player will not get a third eight on either hand. Probability of Blackjack Decks Probability 1 4. I have a very ugly subroutine full of long formulas I determine using probability trees.{/INSERTKEYS}{/PARAGRAPH} For the non-card counter it may be assumed that the odds are the same in each new round. Determine the probability that the player will resplit to 3 hands. Your question however could be rephrased as, "what is the value of the ace, given that the other card is not a ten. It may also be the result of progressive betting or mistakes in strategy. 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. So, the best card for the player is the ace and the best for the dealer is the 5. In that case, the probability of a win, given a resolved bet, is The probability of winning n hands is a row is 0. 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. Any basic statistics book should have a standard normal table which will give the Z statistic of 0. Following this rule will result in an extra unit once every hands. However if you were going to cheat it would be much better to remove an ace, which increases the house edge by 0. What you have experienced is likely the result of some very bad losing streaks. 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. 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. 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. Blackjack is not entirely a game of independent trials like roulette, but the deck is not predisposed to run in streaks. You are forgetting that there are two possible orders, either the ace or the ten can be first. What is important is that you play your cards right. From my blackjack appendix 7 we see that each 9 removed from a single deck game increases the house edge by 0. Since this question was submitted, a player held the dice for rolls on May 23, in Atlantic City. Streaks, such as the dealer drawing a 5 to a 16, are inevitable but not predictable. I would have to do a computer simulation to consider all the other combinations. Add values from steps 4, 8, and The hardest part of all this is step 3. Steve from Phoenix, AZ. I recently replaced my blackjack appendix 4 with some information about the standard deviation which may help. Determine the probability that the player will resplit to 4 hands. 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. Repeat step 3 but multiply by 3 instead of 2. Let n be the number of decks. This is not even a marginal play. 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. I have no problem with increasing your bet when you get a lucky feeling. If I'm playing for fun then I leave the table when I'm not having fun any longer. It would take about 5 years playing blackjack 40 hours a week before this piece of advice saved the player one unit. The fewer the decks and the greater the number of cards the more this is true. 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. However there are other ways you get four aces in the same hand, for example the last card might be an 8 or 9. It is more a matter of degree, the more you play the more your results will approach the house edge. The best play for a billion hands is the best play for one hand. There are cards remaining in the two decks and 32 are tens. You ask a good question for which there is no firm answer. These expected values consider all the numerous ways the hand can play out. 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. According to my blackjack appendix 9H the expected return of standing is So my hitting you will save 6. 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 following table displays the results. 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. 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. In general the variation in the mean is inversely proportional to the square root of the number of hands you play. To test the most likely case to favor hitting, 8 decks and only 3 cards, I ran every possible situation through my combinatorial program. If you were to add a card as the dealer you should add a 5, which increases the house edge by 0. {PARAGRAPH}{INSERTKEYS}This is a typical question one might encounter in an introductory statistics class. Resplitting up to four hands is allowed.