• Appendices
• Miscellaneous
• External Links

1. Blackjack Odds of Winning As mentioned previously, the best possible house advantage you’ll usually see at blackjack tables is 0.5% or a little bit less, even with a solid understanding of basic blackjack strategy. However, it’s theoretically possible for blackjack to actually favor the player with the right rule sets.
2. The following blackjack, card counting and advantage play information is from Wikipedia.org – the free encyclopedia that anyone can edit. Please use caution with information provided here.

Rules

I overhear a lot of bad gambling advice in the casinos. Perhaps the most frequent is this one, 'The object of blackjack is to get as close to 21 as possible, without going over.' No! The object of blackjack is to beat the dealer. To beat the dealer the player must first not bust (go over 21) and second either outscore the dealer or have the dealer bust. Here are the full rules of the game.

1. Blackjack may be played with one to eight decks of 52-card decks.
2. Aces may be counted as 1 or 11 points, 2 to 9 according to pip value, and tens and face cards count as ten points.
3. The value of a hand is the sum of the point values of the individual cards. Except, a 'blackjack' is the highest hand, consisting of an ace and any 10-point card, and it outranks all other 21-point hands.
4. After the players have bet, the dealer will give two cards to each player and two cards to himself. One of the dealer cards is dealt face up. The facedown card is called the 'hole card.'
5. If the dealer has an ace showing, he will offer a side bet called 'insurance.' This side wager pays 2 to 1 if the dealer's hole card is any 10-point card. Insurance wagers are optional and may not exceed half the original wager.
6. If the dealer has a ten or an ace showing (after offering insurance with an ace showing), then he will peek at his facedown card to see if he has a blackjack. If he does, then he will turn it over immediately.
7. If the dealer does have a blackjack, then all wagers (except insurance) will lose, unless the player also has a blackjack, which will result in a push. The dealer will resolve insurance wagers at this time.
8. Play begins with the player to the dealer's left. The following are the choices available to the player:
   • Stand: Player stands pat with his cards.
   • Hit: Player draws another card (and more if he wishes). If this card causes the player's total points to exceed 21 (known as 'breaking' or 'busting') then he loses.
   • Double: Player doubles his bet and gets one, and only one, more card.
   • Split: If the player has a pair, or any two 10-point cards, then he may double his bet and separate his cards into two individual hands. The dealer will automatically give each card a second card. Then, the player may hit, stand, or double normally. However, when splitting aces, each ace gets only one card. Sometimes doubling after splitting is not allowed. If the player gets a ten and ace after splitting, then it counts as 21 points, not a blackjack. Usually the player may keep re-splitting up to a total of four hands. Sometimes re-splitting aces is not allowed.
   • Surrender: The player forfeits half his wager, keeping the other half, and does not play out his hand. This option is only available on the initial two cards, and depending on casino rules, sometimes it is not allowed at all.
9. After each player has had his turn, the dealer will turn over his hole card. If the dealer has 16 or less, then he will draw another card. A special situation is when the dealer has an ace and any number of cards totaling six points (known as a 'soft 17'). At some tables, the dealer will also hit a soft 17.
10. If the dealer goes over 21 points, then any player who didn't already bust will win.
11. If the dealer does not bust, then the higher point total between the player and dealer will win.
12. Winning wagers pay even money, except a winning player blackjack usually pays 3 to 2. Some casinos have been short-paying blackjacks, which is a rule strongly in the casino's favor.

The 'Blackjack (21)' gambling spell increases the luck of any Blackjack player in a casino. The natural odds are not in your favor, but with this money spell, you can improve your odds and your take home. Stop letting the Casino cheat you time and time again. Get even or even better – take the casino! The reason casino blackjack tables have a maximum bet is exactly to counter the Martingale strategy. For example, a $10 minimum Table typically has a $500 maximum. Playing the Martingale strategy, if you lose only 6 hands in succession, then you would need to bet above the maximum on the 7th hand in order to keep the cycle going until you win. The odds of being dealt a natural blackjack are merely 4.8%. Following this chart you will see that the most common two card hand, at 38.7%, is a hand totaling 1-16, which is considered a decision hand. Dealer Final Hand Probabilities This blackjack odds chart shows the dealer final hand probability.

Wizard's Simple Strategy

I've been preaching for years that to play blackjack properly requires memorizing the basic strategy. However, after pitching the basic strategy for 20 years, I've learned that few people have the will to memorize it. In my book, Gambling 102, I presented a 'Simple Strategy,' which is seven simple rules to playing blackjack. The cost due to incorrect plays with the Simple Strategy is 0.53%, under liberal Vegas Strip rules.

Ever since my book was published it has bothered me that the cost in errors to my Simple Strategy was too high. So in September 2009 I developed the following 'Wizard's Strategy.' The cost due to imperfect plays is 0.14% only, relative to liberal Vegas Strip rules. That is the cost of one hand for about every 12 hours of play. Compared to the 250 cells in the Basic Strategy, the Wizard's Strategy has only 21, as follows.

Let me be perfectly clear that this strategy is not right 100% of the time. I continue to get Emails saying that when this strategy was used with my practice game, the player was corrected for following it. For example, my simple strategy says to stand on 12 against a 2, when it is mathematically better to hit. If you want to learn a strategy that is correct all the time you should use the appropriate basic strategy for the set of rules you are playing.

Here are some comments of clarification.

• A 'hard' hand is one that either has no aces, or has aces that are forced to count as point, lest the hand bust. A 'soft' hand is one with at least one ace, which may still count as one or eleven points.
• With a hard 10 or 11, double if you have more points than the dealer, treating a dealer ace as 11 points. Specifically, double with 10 against a 2 to 9, and with 11 against 2 to 10.
• If the strategy says to double, but you have three or more cards, or table rules don't allow soft doubling, then hit, except stand with a soft 18.
• If the strategy says to surrender (16 vs. 10), but you can't for whatever reason, then hit.
• If the strategy says to 'not split,' then treat the hand has a hard total of 8, 10, or 20, according to the pair in question.

A reader named Jeff provided another table of my simple strategy, with exceptions in small print. Details about the Wizard's Simple Strategy can be found in my Blackjack appendix 21.

Basic Strategy

For the appropriate basic strategy for just about any set of rules, please visit my basic strategy calculator. I still have my traditional charts too:

House Edge

See my Blackjack House Edge Calculator to determine the house edge under 6,912 possible rule combinations.

Rule Surveys

Las Vegas: I'm proud to feature up date blackjack rules for every casino in Las Vegas. The list is updated monthly, based on Stanford Wong's Current Blackjack Newsletter. Effective November 2009 the survey has been moved to my companion site, WizardOfVegas.com.

Rule Variations

Following is a list of some common rule variations and the effect on the player's expected return compared to standard U.S. rules (8 decks, dealer stands on soft 17, double after split allowed).

Rule Variations

Rule	Effect
Single deck	0.48%
Early surrender against ten	0.24%
Player may double on any number of cards	0.23%
Double deck	0.19%
Player may draw to split aces	0.19%
Six-card Charlie	0.16%
Player may resplit aces	0.08%
Late surrender	0.08%
Four decks	0.06%
Five decks	0.03%
Six decks	0.02%
Split to only 3 hands	-0.01%
Player may double on 9-11 only	-0.09%
Split to only 2 hands	-0.10%
European no hole card	-0.11%
Player may not double after splitting	-0.14%
Player may double on 10,11 only	-0.18%
Dealer hits on soft 17	-0.22%
Blackjack pays 7-5	-0.45%
Blackjack pays 6-5	-1.39%
Blackjacks pay 1 to 1	-2.27%

I also have a longer list of rule variations.

Beware Short Pays on a Blackjack

More and more tables are showing up that pay less than the full 3 to 2 on a blackjack. Most of these tables pay 6 to 5, but some even money and 7 to 5 tables are known to exist. I would estimate that 10% of '21' tables in Las Vegas now pay less than 3 to 2. In my opinion, only games that pay 3 to 2 deserve to be called 'blackjack,' the rest fall under '21' games, including Super Fun 21 and Spanish 21. Regardless of the other rules, you should demand nothing less than 3 to 2 blackjack. You should always check the felt to be sure, and if the felt doesn't say, look for a sign. If nothing says the win on a blackjack, then ask.

Articles about 6-5 Blackjack:

• Taking a hit: New blackjack odds further tilt advantage toward the house, Las Vegas Sun, Nov. 13, 2003.
• Tighter blackjack rules would hurt players' bankroll, Pittsburgh Tribune-Review, Feb. 4, 2011.

Bad Strategies

Three popular bad strategies encountered at the blackjack table are: never bust, mimic the dealer, and always assume the dealer has a ten in the hole. All three are very bad strategies. Following are my specific comments on each of them, including the house edge under Atlantic City rules (dealer stands on soft 17, split up to 4 hands, double after split, double any two cards) of 0.43%.

Never bust: For my analysis of this strategy I assumed the player would never hit a hard 12 or more. All other decisions were according to correct basic strategy. This 'never bust' strategy results in a house edge of 3.91%.

Mimic the dealer: For my analysis of this strategy I assumed the player would always hit 16 or less and stand on17 or more, including a soft 17. The player never doubled or split, since the dealer is not allowed to do so. This 'mimic the dealer' strategy results in a house edge of 5.48%.

Assume a ten in the hole: For this strategy I first figured out the optimal basic strategy under this assumption. If the dealer had an ace up, then I reverted to proper basic strategy, because the dealer would have peeked for blackjack, making a 10 impossible. This 'assume a ten' strategy results in a house edge of 10.03%.

Play Blackjack

Practice your blackjack game using my two training tools.

Practice Basis Strategy

Practice Card Counting

Written by: Michael Shackleford

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.

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
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 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:

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
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:

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:

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
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:

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?

• 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.

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”.

Blackjack Odds With 16

• 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.)

The probability of drawing those 2 cards in that order is
1/52 X 1/51, or 1/2652.

Example 2:

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
about 5%, which is about 1 in 20.

Example 3:

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.

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.

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.

• 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.

• 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.

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.

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.

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.

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%

Blackjack Odds While Counting

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.

Blackjack Odds Wikipedia

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.

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%

Poker Odds Wikipedia

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.