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Blackjack Math - The Mathematics Behind Advantage Play
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That is because most casino games by nature have a negative expectation for the player.
This means that for every wagered that is made on the game; machine or table game, gives back some amount less than the wagered amount over time.
If 1 million players wager 1 dollar and, one player wins 500K than the casino makes a profit of 500K and an average loss of 50 cents per wager is perceived.
In slot machines the advertised pay back is often in the neighborhood of 97-99%.
This is over the entire life of the machine where a machine may collect 100s of millions of dollars in action over its lifetime.
Table games are slightly different because some include a skill component and the % advantage the casino has varies from player to player.
But the same general principle applies.
This article is an in depth analysis of the math blackjack probability of casino gaming.
The information presented here is valid for play as well as online play.
However; the Blackjack software programs that online casinos use include all of the cards in every new round of play.
The analysis will apply to the game of Blackjack.
Blackjack is a game of dynamic probabilities and shifting percentages.
But even though the percents are constantly changing, the cumulative percentage of the overall advantage remains constant.
This is achieved by taking the sum of the advantages over all possibilities.
For example, if one hand total has an advantage of positive 5% and another hand has a advantage of -6%, than the total advantage for the two hands is +1%.
When the reader understands this game it will be easy to translate the concepts to any other casino game with a static advantage over the player.
GAMING STATISTICS Understanding the statistics involved in casino gaming is essential in evaluating the results.
This assertion is valid for both the player and the casinos.
The knowledge presented here is required to determine whether the results good or bad, lye in the statistical realm of possibility.
This is easily displayed in and Craps.
For example, when a coin is flipped there is a 50% chance that the outcome would be heads and a 50% chance that the outcome would be tails.
If the coin comes up 10 heads in a row the next flip would again have mohegan sun high roller blackjack 50% chance of coming up heads.
In blackjack what happens in the past directly affects what happens in the future.
Blackjack has memory, and the law of independent trials is not valid.
HOW BLACKJACK IS DIFFERENT?
In Blackjack each card has a specific value that it adds to, or subtracts from the initial advantage that the casino has over the player.
The initial advantage is derived from the rules of the game.
When enough of the right cards are dealt, the advantage swings in the players favor.
In blackjack when an Ace or 10 value card is dealt the casino advantage over the player increases.
When lower value cards are put in play continue reading the casino advantage decreases, and when enough of those cards are dealt, the player has an advantage over the casino.
The percent advantage that the casino has over the player in blackjack or vice versa is not static.
There are many approaches that one can introduce to keep track of the shifting percentages.
This system assigns values of either: 1, -1 or 0 to the cards.
All cards 2-6 are assigned a value of 1 and all cards with a face value of math blackjack probability, 8 and 9 have a value of 0.
All tens, face cards and Aces have a value of -1.
As the cards are dealt, the player adds the assigned values of the cards up, the summation of these cards after a round of blackjack is termed the running count.
In a positive running count, the value is normalized into an average of how many more high cards than low cards or low cards than high cards there are per deck.
To accomplish this, the player estimates how many decks are remaining and, the running count is then divided by how many decks remain, and this value is termed the true count.
For example, if a player has observed three math blackjack probability of a link deck shoe being played, and the running count is a 15, that is fifteen more low cards 2-6 have been played than high cards 10s, face cards and aces through the first three decks of the shoe; the player then takes the running count 15 and divides by the decks remaining 3and this would give a true count of 5.
The player subtracts an offset: usually 1, which takes into account the casinos advantage at the start of the deck or shoe this offset is dependent upon several factors such as math blackjack probability rules of the game and the number of decks used and that number, is the number of units the player would wager on the next hand.
For every whole unit increment plus or minus observed in the true count, the player advantage increases by approximately 0.
When a preponderance of high cards remain, the true count is high and the player has an advantage over the casino.
This occurs for three reasons.
First, blackjacks are dealt more frequently and, since the payoff on a blackjack is asymmetric the player gets paid 3:2 on a player blackjack, but only loses his initial bet on a dealer blackjackthis benefits the player.
Usually a player would like to see a high card come out when doubling down or splitting, or the player exercises these options when the dealer is weak and a high card will cause the dealer to break a hit that would cause the dealer to go over 21.
These plays have a higher return when the remaining deck is rich in high cards.
Finally, the player may vary their strategy depending upon the composition of the remaining cards.
With a preponderance of high cards, the https://yournaughtystory.com/blackjack/william-hill-live-blackjack-online.html can stand on more stiff hands totals of 12-16double down more often with strong totals cards equal to 9, 10 or 11 or, when the dealer is weak and susceptible to going over 21, the player may stand.
In contrast, the rules prohibit the dealer from varying their strategy.
The combination of these factors gives rise to situations where the is overcome and a skilled player has an advantage over the house.
CALCULATING THE WIN To determine what the amount that one expects to win over a given time either the casino or playerthree key pieces of information are required.
Number of Hands or Spins 3.
This leads to a zero sum game.
No winners no losers.
AM I WHERE I SHOULD BE?
When a coin is flipped 100 times the outcome is rarely exactly 50 heads and 50 tails.
Therefore we must introduce the concept of variance per number of events.
Variance is a measure of statistical dispersion.
To stick with the coin flip example, variance helps answer the question of whether or not it would it be surprising if we observed 45 heads out of 100 trials, or if we observed only 5 heads in 100 coin flips.
The answers are no and yes.
Getting only 5 heads in 100 coin flips would virtually prove you were flipping a weighted coin.
Understanding this concept is crucial for evaluating casino gaming results, since proper statistical analysis is required in order to determine if the results good or bad are a function of luck or skill.
It essentially determines whether or not a player or casino is being cheated.
Variance is usually discussed in terms of standard deviations, and that will be the case going forward in this discussion.
Standard deviation is equal to the square root of the variance.
The standard deviation for a series of trials is represented by the Greek letter σ sigma math blackjack probability is equal to the standard deviation of each event multiplied by the square root of the number of events.
In the graphical representation the expected value is indicated by the Greek letter µ and the Standard Deviation is represented by the Greek letter σ.
According to the Gaussian distribution curve, there is just over a 68% chance that the result will be within one standard deviation, plus or minus of the expected value.
There is a just over a 95% chance that the results will be within two standard deviations, plus or minus of the expected value.
There is approximately a 99.
Applying this to the scenario of 100 flips of a visit web page we conclude that the standard deviation for 100 trials is 10 times square root of https://yournaughtystory.com/blackjack/whats-a-blackjack.html the standard deviation for a single trial which is 0.
In the coin flip scenario we expect the 50 of the 100 flips to land math blackjack probability heads and 50 of the 100 to land on tails.
Including the standard deviation concept of plus or minus 5, there is a 68% chance that for a 100 flips of a coin the heads side will come up between 45 and 55 times.
Applying the expected value and standard deviation equations to the betting unit of 100 dollars for a casino game with a 1% advantage over the player the following results are computed.
As the number of events increase, the standard deviation gets smaller and smaller mlife blackjack comps to the expected value.
At some point along the curve simple rules of blackjack expected value and standard deviations intersect.
At this point there is an 84% chance that the standard deviation will be less than the expected value.
This means there is an 84% chance that a profit will be made from that point forward and that your funds will never be depleted.
This intersection point for a 1% advantage is shown in the following graph.
FOR SIMPLICITY THE STANDARD DEVIATION VALUE IS ABSOLUTE The intersect point between the expected value and standard deviation is just below 12,000 read more />At 12,000 hands there is an 84% chance that the expected value will surpass the negative standard deviation, indicating the player will not zero out their bankroll 84% of math blackjack probability time.
Computing the same graph with 2% advantage the graph shows an equivalence point that is substantially lower.
This makes sense because casinos are playing the game 24 hours a day 7 days a week.
And because almost all players play to a disadvantage the casinos makes more and more money with less and less variance relative to their expected value.
In forth coming articles I will discuss various aspects of attacking casino games for profit.
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statistics - Blackjack Probability - Mathematics Stack Exchange
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Blackjack Odds and Probability
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The Blackjack Strategy Guide

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Blackjack Probability: What do you Need to Know to Have an Edge?
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Is it even possible?
This is a typical question one might encounter in an introductory statistics class.
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.
From my section on the we find the standard deviation in blackjack to be 1.
Any basic statistics book should have a standard normal table which will give the Z statistic of 0.
So the probability of being ahead in your example is about 18%.
I have a few questions regarding blackjack: How often can one expect the dealer to bust and how often can a player expect to win four hands in a row?
When the dealer stands on a soft 17, the dealer will bust about 29.
When the dealer hits on a soft 17, the dealer will bust about 29.
According to mythe probability of a net win is 42.
However, improbable! downstream casino blackjack join we skip ties, the probability is 46.
So, the probability of a four wins in a row is 0.
First of all, I would like to add my name to the growing list of people who love your web site.
Your information is quite valuable to both the beginning and expert gambler, and you present your findings in a pleasant, understandable, and even humorous manner.
I always check out your site before I head to Las Vegas or Lake Tahoe just to remind me how to play smartly.
Anyway, on to my question.
Well, more of an observation: when the dealer pulls a 5 on a 16 for their sixth consecutive win, there's always someone who gets up and leaves the table, muttering that the dealer is a mean cruel heartless soul, and goes in search of a "hotter" table.
But is there any truth in this?
Obviously the dealer is inconsequential to the cards dealt I like to say the dealer link "simply a messenger of the cards" but are streaks in an 8-deck shoe inevitable, and even predictable?
Or is it more like your roulette example, where the odds of each new round are exactly the same?
Thanks once again for your web site.
Thanks for your kind words.
Streaks, such as the dealer drawing a 5 to a 16, are inevitable but not predictable.
Blackjack is not entirely a game of independent trials like roulette, but the deck is not predisposed to run in streaks.
For the non-card counter it may be assumed that the odds are the same in each new round.
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 https://yournaughtystory.com/blackjack/betfair-blackjack-demo.html next time as the dealer who had been busting on 16 for several hours.
According to mythe probability of an overall win in blackjack is 42.
I'm going to assume you wish to ignore ties for purposes of the streak.
In that case, the probability of a win, given a resolved bet, is 46.
The probability of winning n hands is a row is 0.
So the probability of winning six in a row is 0.
Can it actually be true that what I experience has a statistical base?
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 300 chips, it might take hours to double my money my goalyet 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.
The basic strategy flat bettor should have a roughly symmetrical expectation in terms of steep ups and downs, slightly favoring steep downs due to the house edge and a 48% chance of a losing hand compared to 43% chance of winning.
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 with some information about the standard deviation which may help.
For example this table shows that if you play 10,000 hands of blackjack the probability is 90% of finishing within 192 units where you started after subtracting the expected loss due to the house edge.
So in 10,000 hands you are likely to win or lose less than 2% of total money bet due to here variation.
However if we go up to one million hands the probability is 90% of an 0.
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 go here passes have come out, article source even that's a million to one shot figuratively--literally, it's much worse.
I math blackjack probability 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 154 rolls on May 23, 2009 in Atlantic City.
The probability of this is 1 in 5,590,264,072.
For the probability for any number of throws from 1 to 200, please see my.
For how to solve the problem yourself, see my site, problem 204.
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 103 cards remaining in the two decks and 32 are tens.
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 30.
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 read more 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.
So standing is the marginally better play.
Following this rule will result in an extra unit once every 1117910 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 17.
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 the expected return of standing is -0.
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 10 dollar blackjack atlantic city billion hands is the best play for one 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 10.
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.
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 10.
For example in a six deck game the answer would be 1- 0.
If there were a shuffle between hands the probability would increase substantially.
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 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 https://yournaughtystory.com/blackjack/blackjack-21-cheats.html probability of a string of losses in the August 4, 2002 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 173 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 173 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 173 I meant that starting with the next hand the probability of losing 8 in a row is 1 in 173.
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-10, 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 AcesI 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 was very helpful.
Resplitting up to four hands is allowed.
Here is how I did it.
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.
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 math blackjack probability, 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.
The question: I notice from your May 5, 2003 Column that you actually CALCULATE your blackjack odds.
I am a check this out surprised that you were not using your computer to SIMULATE the results.
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 more info 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 math blackjack probability 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: Player 1 0.
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 click at this page 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 17-21 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!
Thanks for the compliment.
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 100-million hand simulation.
Event Probability Dealer has only blackjack 1 in 22 Player doubles or splits 1 in 7.
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.
As I read your analysis of the side bet inam 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 click here deck is played down sometimes your odds get better, and math blackjack probability worse, but in the long run they average out and stay at a 23.
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 15.
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.
If the probability of a dealer bust is 56% with a six, my calculation suggests the odds of this independent event happening six consecutive times is 0.
At they use 8 decks and the dealer stands on a soft 17.
According to mythe 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 math blackjack probability 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?
According to standard BJ rules and perfect basic strategy, how many percent of my DOUBLED DOWN hands should I expect to win, push and lose?
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.
I was playing strict Basic Strategy for New Zealand conditions not counting, CSM in use.
Have you ever heard of such a horror streak?
My calculations estimate the probability of 19 straight losses as 1 chance in about 207,000; 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 we see the following probabilities for each initial hand.
By way of comparison, the probability of being dealt a royal flush in video poker is 1 in 649,740, 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?
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.
The reason you should surrender is half the sevens have already been removed from the deck.
You need another seven to make 21, the only hand that will beat a dealer 20.
So the shortage of sevens lowers the expected value of hitting to under half a bet, making surrender the better play.
In an eight-deck shoe there are 416 cards.
That may seem like a lot, but 16 against a 10 is such a borderline hand that removal of just one card can making standing a better play.
The rule is that for eight or fewer decks if your 16 is composed of three or more cards, and the dealer has a 10, then you should stand.
In a two-card 16 the average points per card is 8, with a 3-card 16 the average is 5.
With more small cards out of the deck in the 3-card hand the remaining deck becomes more large card rich, making hitting more dangerous, swaying the odds in favor of standing.
Thanks for maintaining this web site!
I have a question about a blackjack rule that is applied in Dutch casinos: When being dealt a pair of sevens, a third seven will earn you 2:1 on your bet, regardless if you win the hand or not.
However, this only applies when the sevens have NOT been split.
I know that there are 6 dealer up cards in basic strategy that allow splitting sevens and 7 that do not, so the player should have an edge in this particular situation.
But what are the odds of being dealt 3 sevens in blackjack in the first place?
And if dealt 3 sevens, what are the odds they qualify for the 2:1 pay-out rule, based on a 4 to 6 decks, dealer stands on soft 17 basic strategy chart?
Hope you can figure this one out for me.
Keep up the good work!
I show that rule is worth 0.
Despite the incentive to hit 7,7 against a dealer 2-7, the player should still follow basic strategy and split.
I bourget blackjack ace he should wait because he could get a two, three, four, five, etc.
What do you think?
Or is my friend just a whiner?
Thank you for math blackjack probability time.
Maybe you can take advantage of his complaining by offering to buy his hand for less than the fair 79 cents on the visit web page />Bally Gaming has a single-deck, multi-hand, blackjack game.
The player plays seven hands against a single dealer hand.
There is an interesting rule in that if the game runs out of cards, all click player hands automatically win.
What is the probability of running out of cards?
Can have suggest any strategy changes to run out the deck?
The house edge using total-dependent basic strategy is 2.
I ran texas rules explained 7-player simulation, using total-dependent basic strategy, and the average number of cards used per round was 21.
In almost 190 million rounds played, the most cards ever used was 42, which happened 7 times.
It is my educated opinion that even with computer perfect composition-dependent strategy the player would still realistically never see the last card.
You could cut down the house edge much more using composition-dependent strategy, according to all the cards seen as you click here along.
I wrote a letter of complaint about it to the casino manager, stating in part: I just wanted to express my disappointment in this change, if it is true.
I never had a chance to take advantage of the promotion and doubt I will be able to now.
Also, you have thirty days in which to complete the card.
I hope you understand this is not a task that is unreachable with that much time.
I THANK YOU for your letter.
Hope you can give it a try and win some money!
What is the probability of getting 30 blackjacks in four hours?
According to myblackjack players play about 70 hands per hour.
I assume a blackjack tie still gets a stamp.
The probability of filling the card in 4 hours, assuming 280 hands, is 1 in 30,000 playing one hand at a time.
I suspect any player achieving the goal in four hours was playing at least two hands at a time.
This question was raised and discussed in the forum of my companion site.
On a recent Vegas trip I saw the dealer get a 9-card 21.
The rules were six decks and the dealer stood on soft 17.
What are the odds of that?
The probability of the dealer getting exactly a 9-card 21 under those rules is 1 in 32,178,035.
Here is the probability for various numbers of decks and whether dealer hits or stands on soft 17.
Decks Stand Soft 17 Hit Soft 17 1 1 in 278,315,855 1 in 214,136,889 2 1 in 67,291,581 1 in 41,838,903 4 1 in 38,218,703 1 in https://yournaughtystory.com/blackjack/atlantic-city-blackjacks-football-team.html 6 1 in 32,178,035 1 in 18,980,158 8 1 in 29,749,421 1 in 17,394,420 Assuming six decks and the dealer stands on soft 17, here is the probability of the dealer getting a 21 or a blackjack in the case of two cardsaccording to the total number of cards.
Cards Probability 2 1 in 21 3 1 in 19 4 1 in 56 5 1 in 323 6 1 in 3,034 7 1 in 42,947 8 1 in 929,766 9 1 in 32,178,035 10 1 in 1,986,902,340 11 1 in 270,757,634,011 12 1 in 167,538,705,629,468 Not that you asked, but the next table shows the probability of the dealer making any non-busted hand under the same rules by the number of cards.
Cards Probability 2 1 in 3 3 1 in 4 4 1 in 12 5 1 in 67 6 1 in 622 7 1 in 8,835 8 1 in 193,508 9 1 in 6,782,912 10 1 in 424,460,108 11 1 in 58,597,858,717 12 1 in 36,553,902,750,535 For more discussion about this question, please visit my forum at.
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The mathematics of blackjack: Probabilities Blackjack We first present the probabilities attached to card dealing and initial predictions.
In making this calculus, circumstantial information such math blackjack probability fraudulent dealing is not taken into account as in all situations corresponding to card games.
All probabilities are calculated for cases visit web page one or two decks of cards.
Let us look at the probabilities for a favorable initial hand the first two cards dealt to be achieved.
A good initial hand which you can stay with could be a blackjack or a hand of 20, 19 or 18 points.
The probabilities of events predicted during the game are calculated on the basis of the played cards the cards showing from a certain moment.
This requires counting certain favorable cards showing for the dealer and for the other players, as well as in your own hand.
Any blackjack strategy is based on counting the cards played.
Unlike a baccarat game, where a maximum of three cards are played for each player, many cards could be played at a certain moment, especially when many players are at the table.
Card counting techniques cannot however be applied in online blackjack.
The formula of probability for obtaining a certain favorable value is similar to that for baccarat and depends on the number of decks of math blackjack probability used.
In the case of a 2-deck game, the probability is: Generally speaking, if playing with math blackjack probability decks, the probability of obtaining a card with the value x is: Example of application of the formula: Assume play with one deck, you are the only player at math blackjack probability, you hold Q, 2, 4, A total value 17 and the face up card of the dealer is a 4.
Let us calculate the probability of achieving 21 points receiving a 4.
If we want to calculate the probability of achieving 19, 20 or 21 points, all we must do is total the three probabilities just calculated.
Unlike in baccarat, where fewer cards are played, the number of players is constant twoand the number of gaming situations is very limited, in blackjack, the number of possible playing configurations is in the thousands and, as a practical matter, cannot be entirely covered by tables of values.
Sources A big part of the gaming situations that require a decision, where the total value held is 15, 16, 17, 18, 19 or 20 points, is comprised math blackjack probability tables in the section titled Blackjack of the book PROBABILITY GUIDE TO GAMBLING: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets.
You will also find there other issues of probability-based blackjack strategy.
See the section for details.
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