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Thank you for your time. Also, you have thirty days in which to complete the card. At the time, Federal Express was the most highly financed new company in U. 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? In other projects Wikimedia Commons.

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Online casinos for usa players mac the the street addresses, commenters is The changes to development files. It seems to me that it takes a lot longer to win X number of chips that to lose the same amount I only play blackjack.

For example, if I start with chips, it might take hours to double my money my goal , yet I can lost that number in what seems like almost no time at all. Can this really be true? Also, do you have a rule of thumb about when to leave the table when you are winning? What you have experienced is likely the result of some very bad losing streaks. It may also be the result of progressive betting or mistakes in strategy. If I'm playing for fun then I leave the table when I'm not having fun any longer.

In a six-deck shoe, what is the percentage of times that a blackjack ace face card or ten will come up?

Let n be the number of decks. Still love your site! I always turn to your site when I'm having questions, most of the time I will find the answer but not always. When playing basic strategy blackjack I understand that I will have ups and downs and over the long run I will roughly break even, my question is what is really "over the long run"?

A month, a year, five years? Thanks for the kind words. You ask a good question for which there is no firm answer. It is more a matter of degree, the more you play the more your results will approach the house edge. I recently replaced my blackjack appendix 4 with some information about the standard deviation which may help.

In general the variation in the mean is inversely proportional to the square root of the number of hands you play. All of this assumes flat betting, otherwise the math really gets messy. Please explain how to calculate the probability of a blackjack occurring in a single deck.

Do you have any idea what the "record" is for the most sevens thrown with a pair of fair dice in craps is? I had someone tell me it was 84, but the odds against that many sevens in a row being thrown is so long I'm skeptical. It seems it's more possible that 84 consecutive passes have come out, but even that's a million to one shot figuratively--literally, it's much worse. I tried to look on the Web but have no idea where I would find something like that. Since this question was submitted, a player held the dice for rolls on May 23, in Atlantic City.

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. For how to solve the problem yourself, see my MathProblems. The standard deviation of one hand is 1. If the first card dealt is an ace what is the probability the dealer will have a blackjack?

There are cards remaining in the two decks and 32 are tens. For blackjack, which is the probability to obtain three suited seven in a 6-deck shoe? There are 24 sevens in the shoe. What piece of information am I missing? If the odds of pulling a ten count card out of a deck is about Why do blackjack simulators and blackjack authors state that the odds for a blackjack are 4. What am I missing? You are forgetting that there are two possible orders, either the ace or the ten can be first.

Our local casino hands out promotional coupons, which act as a first-card ace in blackjack. Do you know the overall expectation of having an ace as your first card? Your question however could be rephrased as, "what is the value of the ace, given that the other card is not a ten.

I checked your web site and I could only find appendixes for multiple card hands in 1 and 2 deck games. Is this article correct? The fewer the decks and the greater the number of cards the more this is true. To test the most likely case to favor hitting, 8 decks and only 3 cards, I ran every possible situation through my combinatorial program. The following table displays the results. Expected Values for 3-card 16 Vs. So standing is the marginally better play.

Following this rule will result in an extra unit once every hands. It would take about 5 years playing blackjack 40 hours a week before this piece of advice saved the player one unit. I play 6 deck blackjack in Tunica, MS. The dealer hits on soft It seems only a 10 or face card can beat this and the odds would be in my favor if the dealer draws more than one card. Also, since most strategies are based on millions of calculations done on a computer, I wonder if those of us who will never play a million hands can rely on slight variations like this one.

Is this a poor, fair or bad move to make? According to my blackjack appendix 9H the expected return of standing is So my hitting you will save 6. This is not even a marginal play. There is no sound bite answer to explain why you should hit. These expected values consider all the numerous ways the hand can play out. The best play for a billion hands is the best play for one hand. If you want to deviate from the basic strategy here are some borderline plays: Deviating on these hands will cost you much less.

My friend and I are debating two blackjack issues that arose from his Caribbean Vacation. House favor or player favor? It depends on the number of decks. Here is the exact answer for various numbers of decks.

Probability of Blackjack Decks Probability 1 4. What is the probability that you play ten hands and never obtain a two-card 21? Assume the cards are reshuffled after each 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. What are the odds of getting 3 blackjacks in a row with 1 deck 4 players and one dealer. If there were a shuffle between hands the probability would increase substantially. In two handed blackjack using one deck, what is the probability of the dealer having a blackjack?

What are the odds of a dealer getting 3 blackjacks in a row on a single deck table with two players? It depends whether there is a shuffle between the blackjacks. Dear Wizard, I was recently playing blackjack with somewhat of a card-shark who also happens to be my friend. We played casino rules, with one deck- and switched the deal after each time the deck expired.

Later, while I was shuffling- I noticed two 9 of spades side by side. My friend obviously claimed he did not know about this, but it seems unlikely. My question is, if you were playing in a similar scenario and were to add one card to the deck, which card would be most advantageous if only you knew about it.

Thank you for your time. From my blackjack appendix 7 we see that each 9 removed from a single deck game increases the house edge by 0.

However if you were going to cheat it would be much better to remove an ace, which 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 best card for the player is the ace and the best for the dealer is the 5.

Occasionally I will increase the bet because I "feel" like I am going to win the next one. I would think that just about all recreational players bet on feel once in a while at least. I was reading through some of your past Ask the Wizard columns and saw your calculation of the probability of a string of losses in the August 4, Column. My question though is what does that really mean? 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?

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? I know, I know, its some sort of divine intervention betting system I am talking about and no betting system affects the house edge. 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 have no problem with increasing your bet when you get a lucky feeling.

What is important is that you play your cards right. Unless you are counting cards you have the free will to bet as much as you want. 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. 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.

I hope this answers your question. Dear wiz, I am a blackjack dealer here in Vegas and the other night dealing, I had 4 out of the 6 ace of spades in my hand.

I had A-A-K-A-A, so good think is I busted, but quick calculations on the game, we figured getting 4 out of the six aces on one had is around 7mil to 1.

Is this number a little high? 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.

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