- You will lose a hand of blackjack well over 50% of the time. Not just a little bit. The reason for that is because, if both you and the dealer bust, you still lose.
- These true odds bets reduce the house's edge in craps to less that 1 percent. Next to blackjack played with extreme discipline, this is the best bet you'll have in casino. The odds on slot.
- When they believed they would get bigger value cards, they started to bet more money as the odds were in their favor. Counting cards is the most well-known technique to win at blackjack, but it can get you thrown out of the casino or even spend a night in jail (as many members of the MIT Blackjack Team found out!). But counting cards is not the.
Blackjack is a slightly deceptive game. Its simple rules of play may fool you into believing it is easy to master but if you delve deeper, you will quickly find this is a purely mathematical game that is all about odds and probabilities. Blackjack Hall of Fame inductees Edward Thorp and Julian Braun were among the first people to come to this realization in the 1960s. They ran millions of simulations on old IBM computers to refine the basic blackjack strategy Ed Thorp published in his Beat the Dealer book, which is now a classic in the blackjack canon.
The importance of odds. The key to becoming a successful blackjack player involves understanding the odds of winning, as well as considering the probabilities of blackjack. No matter what cards you have been given and what situation you face at the table, it’s incredibly important to be aware of the chance of winning and probability of busting.
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If you truly want to win at this game, you need to gain a good understanding of what the odds and probabilities for every possible scenario at the table are and base your playing decisions on these odds. The following article aims at introducing you to the foundations of blackjack odds and probabilities. Toward the end, we have also included several charts that may prove to be useful.
Understanding Blackjack Probabilities
Many people use “probability” and “odds” as two interchangeable terms but in fact, there is a pronounced difference between the two. While inherent in gambling, probability is, first and foremost, a separate branch of mathematics that deals with the likelihood of different events occurring. Probability permeates all aspects of our lives, from weather reports to statistics and playing at your local casino.
Probability is calculated on the basis of known data but cannot be used to predict exact outcomes, like the outcome of a hand in the game of blackjack, for example. It simply shows you the likelihood of an event happening, based on the knowledge of the number of desired outcomes and the number of all possible outcomes. You can use this knowledge to execute the best play at the blackjack table but it alone does not show you with absolute certainty which card the dealer will pull out next.
Statisticians use what is known as a “probability line” to represent the probabilities of events, which can be classified as certain, likely, unlikely, and altogether impossible. The further to the left an event is positioned on the probability line, the more unlikely it is to take place. Conversely, when an event is positioned further to the right of the line’s center, it has a higher likelihood of occurrence.
The probability of a given outcome happening is quite simple to calculate. All you have to do is divide the number of desired outcomes by the number of all possible outcomes. In the context of gambling, this translates into dividing the number of ways to win by the number of all possible outcomes.
Independent vs. Dependent Trials
Before we proceed with concrete examples, we would like to make a distinction between independent and dependent events (or trials in statistics). An independent event has no impact on another event’s probability of occurring (or not occurring). Such is the case with the dice tosses in the game of craps and roulette spins, where previous outcomes have no influence on the results of the trials that are to follow.
Here is an example of determining the probability of rolling a 2 with a six-sided dice. There is only one possible way for you to roll a 2 out of 6 possible outcomes. It follows that the likelihood of a 2 being rolled is 1/6 = 1.166 * 100 = 16.66%.
The probability of rolling a deuce with two dices is even slimmer because there are more permutations (36 to be precise) but there is only one two-dice combination that results in a 2. Respectively, the likelihood of this independent event occurring is 1/36 = 0.027 * 100 = 2.77%. No matter how many times you toss the dice, the probabilities of the tosses, or trials, will always remain the same.
With dependent events, on the other hand, previous trials influence the probabilities of the trials that are to follow. Unlike roulette and dice games, blackjack is a game of dependent trials where each card dealt changes the composition of the remaining deck and therefore, influences the likelihood of forming specific hands on the next rounds of play.
This phenomenon is best explained through examples, so let’s calculate the probability of drawing an Ace from a single deck of cards. Single-deck blackjack utilizes a full deck of 52 cards without any jokers where we have 13 card denominations of 4 different suits each, which is to say there are only 4 Aces out of 52 cards. Therefore, the likelihood of you pulling an Ace at random is P(Ace) = 4/52 = 0.0769 * 100 = 7.69%.
Independent vs. Dependent Trials Additional TipsProvided that the Ace you have already drawn is discarded instead of being reintroduced back into the pack, the probability of pulling an Ace on the next trial will be slimmer. We have three Aces left and the number of cards in the deck has now dropped to 51. The calculations are as follows: P(Ace) = 3/51 = 0.0588 * 100 = 5.88%. The more cards of a given denomination have left the deck, the slimmer the likelihood of drawing a card of said denomination on your next try.
The only unknown factor in the game of blackjack is which card we will pull out next. We can determine the probability of drawing a specific card but cannot say with absolute certainty which card will leave the pack on the next draw.
The only random factor that impacts the draws is the reshuffle. If we place the Ace back into the deck and reshuffle after each trial, the probability of pulling it will remain the same as now you will be dealing with independent trials.
Another Example – Hard 16 vs. Dealer 10 (No Surrender)
Let’s demonstrate how probabilities in blackjack work when more cards have left the deck. We assume you start a fresh round in a no-hole-card game after the single deck has been reshuffled. You are dealt Q-6 against a dealer K but surrender is unavailable, in which case you are forced to hit your hard 16. What is the likelihood of improving your total on the next hit?
We are left with 49 cards since 3 cards have been removed from the deck already. The following cards can help you improve your 16: an Ace for a total of 17 (your Ace will be counted as 1 in this case since otherwise it will bust your hand), a deuce for a total of 18, a 3 for a total of 19, a 4 for a total of 20, and a 5 for the best possible outcome of 21.
Thus, you have 20 cards that can help you out of 49. The probability of drawing a “good” card is 20/49 = 0.408 * 100 = 40.8%. Conversely, the likelihood of you busting by pulling one of the 29 “bad” cardsis 29/49 = 0.591 * 100 = 59.1%.
The Probability of Obtaining a Blackjack
Naturals are the strongest hands you can obtain in the game of 21. Not only it is impossible to lose with a natural (in the worst-case scenario you will push with the dealer) but you get a little extra in terms of profits since blackjacks return 1.5 times your original bet (on condition you are not foolish enough to play 6-to-5 games). Because of this, it is important for you to gain a proper understanding of the probabilities of getting blackjacks.
Knowing the number of decks in play, you can easily determine the likelihood of receiving a natural after the reshuffle. For the purpose, you must multiply the probability of pulling an Ace by the probability of pulling ten-valued cards like 10, J, Q, and K (there are four of each in a single pack for a total of 16). It is also necessary to multiply the result by 2 because there are two possible permutations of cards in a hand of blackjack, for example A-Q and Q-A, K-A and A-K, and so on.
The probability of drawing an Ace is 4/52 while that of pulling one of the ten-value cards is 16/51. The number of cards has dropped to 51 in the second case to account for the Ace that has been removed from the deck. Therefore, we calculate the probability of getting dealt a blackjack in the following way: P (Ace) * P (Ten-Value Card) * 2 = (4/52) * (16/51) * 2 = 0.0482 * 100 = 4.82%.
The likelihood of getting naturals decreases as more decks are introduced into the game, which, inturn, slightly increases the advantage the casino has over you. This often sounds antithetical to inexperienced players who reason it should be the other way around because there are more Aces and ten-value cards when multiple decks are used.
The Probability of Obtaining a Blackjack Additional TipsThis line of reasoning is erroneous because the effect of the individual cards’ removal is not as pronounced in multiple deck games as it is in single or double deck variations. The probability formula we use is the same, however, no matter how many decks are in play.
Below are the probabilities of drawing a blackjack at the start of a fresh shoe with two, four and six decks. You will notice the difference becomes less pronounced the more decks are introduced into play. The difference in blackjack probabilities between six and eight decks is so minuscule, we did not bother including the calculations here.
- The probability of a blackjack with two decks is (8/104) * (32/103) * 2 = 0.0479 * 100 = 4.77%
- The probability of a blackjack with four decks is (16/208) * (64/207) * 2 = 0.0475 * 100 = 4.75%
- 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. In the context of gambling, this corresponds to the ratio between winning and losing outcomes. Unlike probability, the odds are normally expressed as fractions instead of as percentages.
Here are a couple of examples so you can get a firmer grasp on how odds work. Let’s suppose you are interested to know the odds of you hitting number 9 in single-zero roulette where there are 37 numbers on the wheel in total. There is only one number that wins, so it follows there are 36 ways for you to lose. Respectively, the odds for you succeeding are 1 to 36, or 1/36. This corresponds to implied probability of 2.70% which weirdly enough coincides with the advantage of the casino in this game.
Let’s use another example with a single-deck blackjack game. What are the odds of you pulling the Queen of Spades from the 52-card pack? There is only one Queen of Spades in the deck opposed to 51 cards of different suits and denominations, so the odds of you drawing this card are 1 to 51 or 1/51.
In gambling, odds are normally expressed in reverse showing you the chances “against” an outcome occurring, like so: 51 to 1 and 36 to 1. You can convert the implied probability into odds with the following formula: (100/P) – 1 where P stands for probability.
Converting Probability into Odds Additional TipsIn the example with you hitting number 9 on roulette, the calculations run as follows: (100/2.70) – 1 = 37.03 – 1 = 36.03, or roughly 36 to 1. In the one with the Queen of Spades, the implied probability of 7.69%, when converted into “odds against”, corresponds to (100/7.69) – 1 = 13 – 1 = 12, or 12 to 1.
The above calculations show us the actual, true odds of hitting a 9 and of drawing the Queen of Spades from a full 52-card deck on the first trial. The casino extracts its advantage (and profits) by ensuring it always retains a percentage of all players’ cumulative wagers.
In games like roulette and craps, this is achieved solely through payout reduction. The true odds of hitting an individual number on a single-zero wheel are roughly 36 to 1 whereas the casino pays you only 35 to 1.In blackjack, the house extracts its edge in a variety of ways including payout reduction for naturals (6 to 5 instead of 3 to 2), unfavorable rules, and increasing the number of decks in play.
The Probability of the Dealer Busting and the Effect of Cards’ Removal
In blackjack, the odds and probabilities fluctuate with each card that leaves the deck or shoe. This is so because small cards 2 through 6 favor the dealer, whereas high cards 10, J, Q, K, and A favor the player. Cards 7 through 9 are neutral because they favor neither the player nor the dealer.
The dealer has higher chances of exceeding 21 when they start their hand with small cards like 4, 5, and 6. The player’s advantage increases when the dealer exposes one of these cards. Respectively, the player’s advantage begins to drop when high cards are removed from the deck. Examine the table below for more information on the dealer’s probability of busting with individual upcards.
The Dealer’s Exposed Card | The Probability of the Dealer Busting with This Card in S17 Games | The Player Advantage against a Dealer Showing the Card |
---|---|---|
Ace | 11.65% | -16.00% |
2 | 35.30% | 9.80% |
3 | 37.56% | 13.40% |
4 | 40.28% | 18.00% |
5 | 42.89% | 23.20% |
6 | 42.08% | 23.90% |
7 | 25.99% | 14.30% |
8 | 23.86% | 5.40% |
9 | 23.34% | -4.30% |
10, J, Q, K | 21.43% | -16.90% |
Blackjack is the only casino game where cards “have a memory” since your chances of winning change each time a card is removed from the deck. In fact, this is the basic premise of card counting which we discuss further on in this guide.
When the composition of the deck or shoe is such that ten-value cards and Aces outnumber small card, the player holds an advantage over the dealer. It is the other way around when there are more small cards left to be played. The table below shows you how the cards of different ranks impact your chances of winning:
Card That Leaves the Deck | Impact of the Card’s Removal on Players’ Chances of Winning |
---|---|
A | -0.59% |
2 | 0.40% |
3 | 0.43% |
4 | 0.52% |
5 | 0.67% |
6 | 0.45% |
7 | 0.30% |
8 | 0.01% |
9 | -0.15 |
10, J, Q, K | -0.51 |
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.
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.
This page takes a look at how blackjack probability works. It
also includes sections on the odds in various blackjack
situations you might encounter.
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. Free pch slots.
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. Free pch slots.
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.
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.
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
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.
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
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.
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.
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.
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
single die. There is no seven, so there are zero ways for this
to happen out of six possible results. 0/6 = 0.
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
deck of cards with no joker in it. There are zero jokers and 52
possible cards to draw. 0/52 = 0.
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.
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%.
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.
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.
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
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.
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.
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.
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
Pirate ship slot. gets to guess a number. If he’s right, you pay him $5. If he’s
wrong, he pays you $1.
made up. He bets a dollar on every roll of a single die, and he
Pirate ship slot. 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.
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.
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?
game at a profit. Suppose you only paid him $4 every time he
won. You’d have him at an advantage, wouldn’t you?
- He’d win an average of $4 once every six rolls
- But he’d lose an average of $5 on every six rolls
- This gives him a net loss of $1 for every six rolls.
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.
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.
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.
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.
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”.
The operative words in these situations are “and” and “or”.
- If you want to know the probability of A happening AND
of B happening, you multiply the probabilities. - If you want to know the probability of A happening OR of
B happening, you add the probabilities together.
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.)
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
1/52 X 1/51, or 1/2652.
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.
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.
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.
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.
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.
1/13 X 16/51, or 16/663.
The probability of being dealt a 10 and then an ace is also
16/663.
16/663.
You want to know if one or the other is going to happen, so
you add the two probabilities together.
you add the two probabilities together.
16/663 + 16/663 = 32/663.
Classic slots app. That translates to approximately 0.0483, or 4.83%. That’s
about 5%, which is about 1 in 20.
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.
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.)
(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.
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.
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.
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.
you’ll raise your bet in these situations.
![What Are The Odds Of Winning At Blackjack What Are The Odds Of Winning At Blackjack](https://i.pinimg.com/originals/7a/a7/2a/7aa72a07100070b2ec2cecc92d3f3388.gif)
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.
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%.
5%.
Expected value is just the average amount of money you’ll win
or lose on a bet over a huge number of trials.
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.
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.
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.
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.
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.
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.
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%.
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.
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.
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.
- You are a perfect basic strategy player in a game with a
0.5% house edge. - You’re playing for $100 per hand, and you’re averaging
50 hands per hour. - You’re putting $5,000 into action each hour ($100 x 50).
- 0.5% of $5,000 is $25.
- You’re expected (mathematically) to lose $25 per hour.
Here’s another example that assumes you’re a skilled card
counter.
counter.
- You’re able to count cards well enough to get a 1% edge
over the casino. - You’re playing the same 50 hands per hour at $100 per
hand. - Again, you’re putting $5,000 into action each hour ($100
x $50). - 1% of $5,000 is $50.
- Now, instead of losing $25/hour, you’re winning $50 per
hour.
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.
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.
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.
conditions.
Rules Variation | Effect on House Edge |
---|---|
6 to 5 payout on a natural instead of the stand 3 to 2 payout | +1.3% |
Not having the option to surrender | +0.08% |
8 decks instead of 1 deck | +0.61% |
Dealer hits a soft 17 instead of standing | +0.21% |
Player is not allowed to double after splitting | +0.14% |
Player is only allowed to double with a total of 10 or 11 | +0.18% |
Player isn’t allowed to re-split aces | +0.07% |
Player isn’t allow to hit split aces | +0.18% |
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.
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.
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?
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.
a certain rank from the deck.
Card Rank | Effect on House Edge When Removed |
---|---|
2 | -0.40% |
3 | -0.43% |
4 | -0.52% |
5 | -0.67% |
6 | -0.45% |
7 | -0.30% |
8 | -0.01% |
9 | +0.15% |
10 | +0.51% |
A | +0.59% |
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.
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.
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.
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.
based on her up card. This provides some insight into how basic
strategy decisions work.
Dealer’s Up Card | Percentage Chance Dealer Will Bust |
---|---|
2 | 35.30% |
3 | 37.56% |
4 | 40.28% |
5 | 42.89% |
6 | 42.08% |
7 | 25.99% |
8 | 23.86% |
9 | 23.34% |
10 | 21.43% |
A | 11.65% |
Odds Of Winning Blackjack At Casino
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.
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
What Are The Odds Of Winning At Blackjack Table
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.
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.