One of the most interesting aspects of blackjack is the
probability math involved. It’s more complicated than other games. In fact, it’s easier for computer programs to calculate blackjack probability by running billions of simulated hands than it is to calculate the massive number of possible outcomes.
Blackjack and Probability Chongwu Ruan Math 190S-Hubert Bray July 24, 2017 1 Introduction Blackjack is an usual game in gambling house and to beat the dealer and make money, people have done lots of research on it. They come up with several basic strategy which is consist of three tables corresponding to the different rules. Blackjack Dealer Outcome Probabilities The dealer outcome probability charts on this page can be helpful in determining which of two tournament strategies is preferable. The values in these charts were calculated using a combinatorial analysis from a full deck after removing only the dealer’s upcard.
Optimal Blackjack Strategy The rules of play 1. The number of players. At most blackjack tables, there is one dealer and from one to six players. The player to the dealer’s left receives his cards rst, and is said to be on rst base.' The nal player on the dealer’s right is the last to get his cards, and is said to be on third base.' The probability of a blackjack with six decks is (24/312). (96/311). 2 = 0.0474. 100 = 4.74% Converting Probability into Odds Odds are different than probability in that they show us the ratio between the number of desired outcomes occurring and the number of ways in which the desired outcome will not occur. Use our sample 'Blackjack Chart.' Read it or download it for free. Free help from wikiHow.
This page takes a look at how blackjack probability works. It
also includes sections on the odds in various blackjack situations you might encounter. An Introduction to Probability
Probability is the branch of mathematics that deals with the
likelihood of events. When a meteorologist estimates a 50% chance of rain on Tuesday, there’s more than meteorology at work. There’s also math.
Probability is also the branch of math that governs gambling.
After all, what is gambling besides placing bets on various events? When you can analyze the payoff of the bet in relation to the odds of winning, you can determine whether or not a bet is a long term winner or loser. The Probability Formula
The basic formula for probability is simple. You divide the
number of ways something can happen by the total possible number of events.
Here are three examples.
Example 1:
You want to determine the probability of getting heads when Example 2:
you flip a coin. You only have one way of getting heads, but there are two possible outcomes—heads or tails. So the probability of getting heads is 1/2.
You want to determine the probability of rolling a 6 on a Example 3:
standard die. You have one possible way of rolling a six, but there are six possible results. Your probability of rolling a six is 1/6.
You want to determine the probability of drawing the ace of
spades out of a deck of cards. There’s only one ace of spades in a deck of cards, but there are 52 cards total. Your probability of drawing the ace of spades is 1/52.
A probability is always a number between 0 and 1. An event
with a probability of 0 will never happen. An event with a probability of 1 will always happen.
Here are three more examples.
Example 4:
You want to know the probability of rolling a seven on a Example 5:
single die. There is no seven, so there are zero ways for this to happen out of six possible results. 0/6 = 0.
You want to know the probability of drawing a joker out of a Example 6:
deck of cards with no joker in it. There are zero jokers and 52 possible cards to draw. 0/52 = 0.
You have a two headed coin. Your probability of getting heads
is 100%. You have two possible outcomes, and both of them are heads, which is 2/2 = 1.
A fraction is just one way of expressing a probability,
though. You can also express fractions as a decimal or a percentage. So 1/2 is the same as 0.5 and 50%.
You probably remember how to convert a fraction into a
decimal or a percentage from junior high school math, though. Expressing a Probability in Odds Format
The more interesting and useful way to express probability is
in odds format. When you’re expressing a probability as odds, you compare the number of ways it can’t happen with the number of ways it can happen.
Here are a couple of examples of this.
Example 1:
You want to express your chances of rolling a six on a six Example 2:
sided die in odds format. There are five ways to get something other than a six, and only one way to get a six, so the odds are 5 to 1.
You want to express the odds of drawing an ace of spades out
a deck of cards. 51 of those cards are something else, but one of those cards is the ace, so the odds are 51 to 1.
Odds become useful when you compare them with payouts on
bets. True odds are when a bet pays off at the same rate as its probability.
Here’s an example of true odds:
You and your buddy are playing a simple gambling game you
made up. He bets a dollar on every roll of a single die, and he gets to guess a number. If he’s right, you pay him $5. If he’s wrong, he pays you $1.
Since the odds of him winning are 5 to 1, and the payoff is
also 5 to 1, you’re playing a game with true odds. In the long run, you’ll both break even. In the short run, of course, anything can happen. Probability and Expected Value
One of the truisms about probability is that the greater the
number of trials, the closer you’ll get to the expected results.
If you changed the equation slightly, you could play this
game at a profit. Suppose you only paid him $4 every time he won. You’d have him at an advantage, wouldn’t you?
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You can reduce that to how much he expects to lose on every
single roll by dividing $1 by 6. You’ll get 16.67 cents.
On the other hand, if you paid him $7 every time he won, he’d
have an advantage over you. He’d still lose more often than he’d win. But his winnings would be large enough to compensate for those 5 losses and then some.
The difference between the payout odds on a bet and the true
odds is where every casino in the world makes its money. The only bet in the casino which offers a true odds payout is the odds bet in craps, and you have to make a bet at a disadvantage before you can place that bet.
Here’s an actual example of how odds work in a casino. A
roulette wheel has 38 numbers on it. Your odds of picking the correct number are therefore 37 to 1. A bet on a single number in roulette only pays off at 35 to 1.
You can also look at the odds of multiple events occurring.
The operative words in these situations are “and” and “or”.
Here are some examples of how that works.
Example 1:
You want to know the probability that you’ll draw an ace of
spades AND then draw the jack of spades. The probability of drawing the ace of spades is 1/52. The probability of then drawing the jack of spades is 1/51. (That’s not a typo—you already drew the ace of spades, so you only have 51 cards left in the deck.)
The probability of drawing those 2 cards in that order is Example 2:
1/52 X 1/51, or 1/2652.
You want to know the probability that you’ll get a blackjack.
That’s easily calculated, but it varies based on how many decks are being used. For this example, we’ll use one deck.
To get a blackjack, you need either an ace-ten combination,
or a ten-ace combination. Order doesn’t matter, because either will have the same chance of happening.
Your probability of getting an ace on your first card is
4/52. You have four aces in the deck, and you have 52 total cards. That reduces down to 1/13.
Your probability of getting a ten on your second card is
16/51. There are 16 cards in the deck with a value of ten; four each of a jack, queen, king, and ten.
So your probability of being dealt an ace and then a 10 is
1/13 X 16/51, or 16/663.
The probability of being dealt a 10 and then an ace is also
16/663.
You want to know if one or the other is going to happen, so
you add the two probabilities together.
16/663 + 16/663 = 32/663.
That translates to approximately 0.0483, or 4.83%. That’s Example 3:
about 5%, which is about 1 in 20.
You’re playing in a single deck blackjack game, and you’ve
seen 4 hands against the dealer. In all 4 of those hands, no ace or 10 has appeared. You’ve seen a total of 24 cards.
What is your probability of getting a blackjack now?
Your probability of getting an ace is now 4/28, or 1/7.
(There are only 28 cards left in the deck.)
Your probability of getting a 10 is now 16/27.
Your probability of getting an ace and then a 10 is 1/7 X
16/27, or 16/189.
Again, you could get a blackjack by getting an ace and a ten
or by getting a ten and then an ace, so you add the two probabilities together.
16/189 + 16/189 = 32/189
Your chance of getting a blackjack is now 16.9%.
This last example demonstrates why counting cards works. The
deck has a memory of sorts. If you track the ratio of aces and tens to the low cards in the deck, you can tell when you’re more likely to be dealt a blackjack.
Since that hand pays out at 3 to 2 instead of even money,
you’ll raise your bet in these situations. The House Edge
The house edge is a related concept. It’s a calculation of
your expected value in relation to the amount of your bet.
Here’s an example.
If the expected value of a $100 bet is $95, the house edge is
5%.
Expected value is just the average amount of money you’ll win
or lose on a bet over a huge number of trials.
Using a simple example from earlier, let’s suppose you are a
12 year old entrepreneur, and you open a small casino on the street corner. You allow your customers to roll a six sided die and guess which result they’ll get. They have to bet a dollar, and they get a $4 win if they’re right with their guess.
Over every six trials, the probability is that you’ll win
five bets and lose one bet. You win $5 and lose $4 for a net win of $1 for every 6 bets.
$1 divided by six bets is 16.67 cents.
Your house edge is 16.67% for this game.
The expected value of that $1 bet, for the customer, is about
84 cents. The expected value of each of those bets–for you–is $1.16.
That’s how the casino does the math on all its casino games,
and the casino makes sure that the house edge is always in their favor.
With blackjack, calculating this house edge is harder. After
all, you have to keep up with the expected value for every situation and then add those together. Luckily, this is easy enough to do with a computer. We’d hate to have to work it out with a pencil and paper, though.
What does the house edge for blackjack amount to, then?
It depends on the game and the rules variations in place. It
also depends on the quality of your decisions. If you play perfectly in every situation—making the move with the highest possible expected value—then the house edge is usually between 0.5% and 1%.
If you just guess at what the correct play is in every
situation, you can add between 2% and 4% to that number. Even for the gambler who ignores basic strategy, blackjack is one of the best games in the casino. Expected Hourly Loss and/or Win
You can use this information to estimate how much money
you’re liable to lose or win per hour in the casino. Of course, this expected hourly win or loss rate is an average over a long period of time. Over any small number of sessions, your results will vary wildly from the expectation.
Here’s an example of how that calculation works.
Here’s another example that assumes you’re a skilled card
counter.
Effects of Different Rules on the House Edge
The conditions under which you play blackjack affect the
house edge. For example, the more decks in play, the higher the house edge. If the dealer hits a soft 17 instead of standing, the house edge goes up. Getting paid 6 to 5 instead of 3 to 2 for a blackjack also increases the house edge.
Luckily, we know the effect each of these changes has on the
house edge. Using this information, we can make educated decisions about which games to play and which games to avoid.
Here’s a table with some of the effects of various rule
conditions.
These are just some examples. There are multiple rules
variations you can find, some of which are so dramatic that the game gets a different name entirely. Examples include Spanish 21 and Double Exposure.
The composition of the deck affects the house edge, too. We
touched on this earlier when discussing how card counting works. But we can go into more detail here.
Every card that is removed from the deck moves the house edge
up or down on the subsequent hands. This might not make sense initially, but think about it. If you removed all the aces from the deck, it would be impossible to get a 3 to 2 payout on a blackjack. That would increase the house edge significantly, wouldn’t it?
Here’s the effect on the house edge when you remove a card of
a certain rank from the deck.
These percentages are based on a single deck. If you’re
playing in a game with multiple decks, the effect of the removal of each card is diluted by the number of decks in play.
Looking at these numbers is telling, especially when you
compare these percentages with the values given to the cards when counting. The low cards (2-6) have the most dramatic effect on the house edge. That’s why almost all counting systems assign a value to each of them. The middle cards (7-9) have a much smaller effect. Then the high cards, aces and tens, also have a large effect.
The most important cards are the aces and the fives. Each of
those cards is worth over 0.5% to the house edge. That’s why the simplest card counting system, the ace-five count, only tracks those two ranks. They’re that powerful.
You can also look at the probability that a dealer will bust
based on her up card. This provides some insight into how basic strategy decisions work.
Perceptive readers will notice a big jump in the probability
of a dealer busting between the numbers six and seven. They’ll also notice a similar division on most basic strategy charts. Players generally stand more often when the dealer has a six or lower showing. That’s because the dealer has a significantly greater chance of going bust. Summary and Further Reading
Odds and probability in blackjack is a subject with endless
ramifications. The most important concepts to understand are how to calculate probability, how to understand expected value, and how to quantify the house edge. Understanding the underlying probabilities in the game makes learning basic strategy and card counting techniques easier. Table Of Contents
Everyone say that playing blackjack is easy — but I disagree.
Blackjack is one of those games that can take years to learn, especially if you want to play well and use all the winning chances you have.
To be good at Blackjack you need to know the right strategy, learn how to play every possible combination of cards, and have a super-sharp memory.
Unless you use a Blackjack cheat card like the one I'm giving away in this article.
Oh, and you need to be lucky too - of course.
The players who get all the chances to win at Blackjack that the game allows, know exactly what hand to play and when they are supposed to Hit or Stand.
Good Blackjack players leave to chance nothing more than they are obliges to. Blackjack is a gambling game and it comes with a natural house edge — but anything more than that, is a mistake Blackjack players can't afford to make.
To be that good, you can either spend a lot of time studying the game and learning all about the best ways to play Blackjack…
…or you can become well-versed in the important aspects of the Blackjack basic strategy with an easy Blackjack cheat sheet for beginners.
The No-Cheat Sheed Approach to Beating Blackjack
If you choose the first approach, you are in for a lot of work that is going to pay out, eventually.
I respect your attitude and I want to help. So here's a list of good resources you should read before you download my printable Blackjack strategy card.
How to Play Blackjack for Beginners. See it as a dummies guide to Blackjack. Here you learn how to count points, understand the action, and move your first steps into the game.
How to Practice Blackjack Online. The best way to learn how to play (and put my Blackjack cheat sheet into practice) is to play for free and train. This article gives you the chance to do just that.
Best Free Blackjack Games to Play for Fun. Need more free games to test the PDF cheat card? This page has everything you need.
Single Deck Blackjack Strategy. Single Deck games are the Eldorado of Blackjack. This article tells you why and gives you some tips to get more winning chances than most other players.
Blackjack Side Bets Explained in Plain English. Lots of people get confused about the side bets in Blackjack — especially as those aren't covered in my cheatsheet. This article tells you which one to try and which ones to avoid.
Best Live Blackjack Casinos in 2019: Updated List. A collection of all the best sites to play Live Dealer Blackjack in 2019 with short reviews of what makes them special.
The No-Cheat Sheet Approach to Beating Blackjack
If you don't want to go through all those articles but you just want to play, I have a way to get you in a position to get better odds at Blackjack.
With my Blackjack Cheat Sheet you:
YES. I am talking about a Blackjack Cheat Sheet pdf for dummies that can help you play like you are supposed to. Download it, print it, memorize it, and use to play your games.
It's often said that in gambling, the house always wins. Unfortunately, that's not a lie.
Although this is true for the majority of players - especially for those who do not know how to play the house effectively - there are ways to increase your chances.
Nobody expects a beginner to win, and you must use this lack of expectations to your advantage.
Blackjack Winning Odds
My Blackjack Cheat Sheet card helps you play Blackjack like players who have been playing the game for years and get more chances.
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Although we call this a Cheat Sheet,it is really cheating? Of course not.
Cheating at the Casino is not allowed. And stupid.
What you find here is a Blackjack strategy that helps you to play better.
Think of it like playing a video game with a guide, or going on a guided tour when you go on vacation.
That's not cheating - that's just sensible precautions. Sure, you think, but isn't blackjack a game, and wouldn't a winning strategy be against the spirit of that game?
If you're thinking that, then, my friend, you are nobler than most players. The world might need more people like you - but you need to change your approach if you want to have your best chances to win at Blackjack.
First, though, let's take you through some basics of deck Blackjack terminology and Blackjack strategy, as you'll need to know what these words and phrases mean in order to decipher what our Cheat Sheet tells you.
Soft hand: This is when there is an ace in your hand, meaning there is no chance of you busting after the draw. The ace can be 1 or 11, but your hand must not total more than 21. Here are those soft combinations:
Blackjack Chart Printable
Hard hand: The combined total of any two cards not containing an ace. Here are the possible hard hands:
It's riskier to draw with a hard hand, as there is more chance of busting.
Hit: drawing another card.
Stand: when your hand is good enough. Double down: this is when you, or another player, increase the wager.
This Cheat Sheet alone won't make you a winner.
This Cheat Sheet helps you to understand the basics and shows you how to play a hand well. It tells you when to take a positive risk and when to hold back.
It's the Yoda you need to become Blackjack's Luke Skywalker.
To win at blackjack, download the cheat sheet, you must.
Can you use the Blackjack Cheat Sheet at the table?
You can keep the easy Blackjack Cheat Sheet open in your browser while a game of online Blackjack is running.
Use it as a reference to understand what decisions you are supposed to make when you play a hand of Blackjack.
After a while, once you get good enough to play on your own, you won't need it anymore. Until that happens, use it while you play. It'll help you learn to make the calls and do the right moves.
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