Single-deck blackjack utilizes a full deck of 52 cards without any jokers where we have 13 card denominations of 4.

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However, the cards in a standard deck are 52 in total which leads to the conclusion that the probability of receiving this card is 1/ Each outcome that can.

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The reason that the house always has an edge on us players in blackjack is the simple fact that we have to act first at the table. Every time we hit a card and bust,.

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Your chance of getting a blackjack is now %. This last example demonstrates why counting cards works. The deck has a memory of sorts. If you track the ratio.

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The probability distribution for X, based on equally likely chances of drawing any particular card is: Pr (X = x) = f (x) = { 16 49, x = 10 2 49, x = 9 3 49, x = 7 4 49.

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There are cards remaining in the two decks and 32 are tens. So the probability of a blackjack is 32/=%. What percentage of hands are suited.

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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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There is only one such card in a single deck and the rest of the remaining 51 cards are not an ace of diamonds for sure. Therefore, the odds in this situation will be.

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The probability distribution for X, based on equally likely chances of drawing any particular card is: Pr (X = x) = f (x) = { 16 49, x = 10 2 49, x = 9 3 49, x = 7 4 49.

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Consider a simplified game in which the deck consists of just two cards: an Ace and a Deuce. The dealer deals just one card per hand and you win if it is an Ace.

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