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 Of Blackjack Single Deck
- Blackjack Probability Chart
- Blackjack Statistics
- Probabilities In Blackjack
This page takes a look at how blackjack probability works. It
also includes sections on the odds in various blackjack
situations you might encounter.
Oct 23, 2007 What is the probability of getting a 'blackjack' hand (21) from drawing the first two cards from a fair deck?.Assume no jokers (52 cards per deck).Assume randomly shuffled. I usually throw in my own assumption of how to go about figuring the question out, and often put my own guess as to the answer, then ask you to check it.but I'm really not sure how to approach this, as the. The probability to get a blackjack (natural): 64 / 1326 =.0483 = 4.83%. 2) Let's do now the calculations for arrangements. (The combinations are also considered boxed arrangements; i.e. The order of the elements does not count).
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.
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.
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.
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.
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.
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:
Probability Of Blackjack Single Deck
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”.
- 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.
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.
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.
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.
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.
- 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% |
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.
| 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% |
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.
Blackjack probability is just like any other probability in the casino. It’s a means of measuring the likelihood of certain events. You’ll often see probabilities expressed as percentages, but they can be expressed as fractions or odds, too.
Blackjack statistics is a way of measuring your actual results and comparing them to your predicted results. In the long run, your actual results will start to resemble your predicted results. But in the short run, random chance will ensure that anything can happen.
That’s why some players have huge losing streaks, while others have big winning streaks. The casino doesn’t worry about this, because they’ve set up the games and the payouts in such a way that they’re ensured a profit in the long run. That’s a matter of expected value.
Some Definitions Related to Probability and Expected Value
In fact, that’s probably the best way to introduce this blog post—with some definitions of some terms related to blackjack probability in general. That way you’ll be able to dig deeper into the main points of the post below.
Let’s start with the phrase “probability.” The word has 2 meanings. The first is that probability is the branch of mathematics that deals with the likelihood of an event occurring. The 2nd is more useful—probability also refers to an event’s likelihood.
Probability is measured numerically, and an event’s 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.
An event with a probability of 50% will happen half the time, on average. 50% is one of the more common ways to express that probability, but you could also say that this event has a probability of 1/2 and still be correct. Another useful way to express probability is in odds format. That’s when you compare the number of ways something can’t happen with the number of ways it can happen. With a 50% probability, an event has “even odds,” or 1 to 1 odds.
Expressing probabilities as odds can be useful when trying to decide whether you have an edge or not. In most casinos, the games all have a built-in edge, but blackjack is exceptional in this respect. I’ll get into that a little later in this post.
Another important concept in gambling probability to understand is the concept of “expected value.” This is what a bet is “worth.” A bet’s expected value can be positive or negative, but if you’re a player in a casino, it’s almost always negative. The formula for expected value is simple, too:
You multiply the probability of winning by the amount you stand to win. You also multiply the probability of losing by the amount you stand to lose. You subtract one from the other, and you have the expected value of the bet.
For example, if you have a 50% chance of winning $1, and you also have a 50% chance of losing $1, you have an expected value of 0. That bet is a break-even bet; over time, you won’t win any money at it or lose any money at it.
But let’s say you have a 45% chance of winning $1, and you have a 55% chance of losing $1. Now your expected value looks quite different:
+$0.45 – $0.55 = -$0.10
This means that over time you’ll lose 10 cents every time you make this bet.
Almost all casino game bets have a negative expected value. You’ll either lose more often than you’ll win, or you’ll win too little when you win to break even, or some combination of these factors. That’s how casinos stay in business.
That’s also why gamblers walk away a winner. In the scenario outlined above, you can’t lose 10 cents on a single bet or even a series of 2 or 3 bets. You’re going to win or lose $1 on each hand.
The expected value is an average expected over the long run.
And the long run is longer than most people think.
That’s why the casino can afford to pay winners occasionally and still make a huge net profit overall.
“The house edge” is another way of looking at the expected value of a bet, but it’s only used to describe bets where the casino has an edge over the player.
How the Casino Wins Consistently at Blackjack
You would think that the casino would have no edge in a game like blackjack. After all, the dealer is getting the same cards as the players. He has the same probability of being dealt a blackjack or going bust as a player.
The amazing thing about the house edge in casino games is that it’s usually a simple byproduct of the rules used by the casino for the game. For example, in roulette, the house gets an edge by paying off all the bets as if the 0 and the 00 weren’t on the wheel.
In blackjack, the house gets its edge by making the players resolve their actions and bets first before the dealer acts. In other words, you must make all your playing decisions before the dealer ever acts. This means that if you bust (get a total of 22 or higher), you automatically lose your bet—even if the dealer also goes bust. Since you acted first, and the dealer resolved your bust before having to play, the house has an advantage.’
This is a huge advantage made bigger by the fact that some players don’t play their hands optimally from a mathematical standpoint. In many cases, the best play is to stand on a hand which isn’t likely to win unless the dealer busts. A lot of players have trouble with this.
This advantage is so big for the casino that it can even afford to offer an extra high payout on some hands. In most casinos, a 2-card hand totaling 21 (a “blackjack” or “natural”) pays off at 3 to 2 odds. This means if you bet $100 and get a blackjack, you win $150.
The casinos can afford this bonus payout and still have a profitable mathematical edge over the player. This 3 to 2 payout is one of the reasons that smart players can get an edge over the casino, and I’ll have more to say about that later in this post.
Since there a finite number of cards in a blackjack deck, it’s possible to calculate the mathematically best play in every situation. This is called “basic strategy.” Computer programs analyze the potential results of every possible decision in every possible situation. The move with the highest expected value is the correct playing decision.
The average blackjack player loses an average of 5% of every bet he places at the blackjack table. The average blackjack player is playing with “common sense,” “hunches,” or just pure dumb instinct.
The smart blackjack player, though, memorizes and uses basic strategy in every situation. This reduces the house edge to less than 1%. Depending on the rules variations in effect at a specific blackjack table, the house edge might be significantly less than 0.5%.
But it doesn’t matter how low the house edge is. If the house has an edge over the player, if the player gambles long enough, he’ll eventually lose all his money. That’s how the casinos stay in business.
Blackjack, though, is different from almost every other game in the casino. It’s a game where a smart player with the right strategy can get an edge over the casino. This is beyond the abilities of most players, and even a lot of players who THINK they’re playing with an edge over the casino are mistaken.
I talk about why and how that is in the next section.
How Probability in Blackjack Differs from Probability in Other Casino Games
The reason a strategic player can get an edge in blackjack is because as each card gets dealt, the composition of the deck as a whole changes. In any random shuffle of a 52-card deck, the cards might fall in any given pattern. But sometimes higher-value cards and lower-value cards are dispersed in the deck unevenly.
When I say “higher-value cards,” I means 10s and aces. Since these are the only cards that can create a blackjack—and the corresponding 3 to 2 payout—it’s better for the player if there are a relatively large number of these cards left in the deck.
“Lower-value cards,” on the other hand, increase your probability of going bust when you take a hit. They also make it harder to hit your 3 to 2 payout on the blackjack. If a deck has a relatively higher percentage of lower-value cards in it, the casino has a bigger edge than usual.
This might seem obvious, but think about it this way if it still isn’t clear:
You’re playing blackjack, and over the course of the 1st couple of hands, all 4 of the aces are dealt.
What is the probability of being dealt a natural after this?
Since you need a 10 AND an ace to get a natural, your probability of getting a natural drops to 0.
Here’s another way to think about it:
When you’re playing roulette or craps, the odds are the same on every outcome. That’s because the number of possible outcomes on a roulette wheel don’t change. You always have 38 numbers with an equal probability of coming up.
When you’re playing craps, those 2 dice have the same number of sides (6) every time you roll them.
You don’t eliminate a number from the roulette wheel once a ball has landed in that slot. You start over on the next spin.
You don’t eliminate a number from the sides of the dice just because it came up on the previous roll.
But when a card gets dealt in blackjack, it’s gone from the deck until the deck gets re-shuffled.
That changes the probabilities on every hand.
How You Can Use This Information to Get an Edge over the Casino in Blackjack
If you could bet more when the deck has a higher ratio of 10s and aces and bet less when it doesn’t, you could get an edge over the casino. You’d be putting more money into action when you’re more likely to get a 3 to 2 payout.
And as it turns out, you CAN do exactly that.
You’ve probably heard of “card counting.”
Unless you’ve read about it before, you probably think it’s beyond the capabilities of most mere mortals. Maybe you saw Rain Man as a teenager and think you need to be able to memorize every card as it’s played to succeed in counting cards.
But the truth is, anyone who can add and subtract 1 can count cards. Maintaining the level of concentration of keeping that count accurate while not looking like you’re counting is the real trick.
You don’t track specific cards when you’re counting cards in blackjack. You just track the ratio of high cards to low cards. You assign a value of -1 to the 10s and aces, for example. Then you assign a value of +1 to the 2s, 3s, 4s, 5s, and 6s. The 7, 8, and 9 have a 0 value each.
When the running count is positive you bet more. The higher the count, the more you bet.
When the running count is 0, you bet less.
By doing this, you get a mathematical edge over the casino.
You can also use the count to inform your basic strategy decisions. Basic strategy assumes a full 52-card deck, but once the ratios of cards change as the deck gets depleted, the playing decision with the highest expected value can change.
This can increase your edge over the house even further.
You don’t gain a huge edge over the casino through card counting, though. You usually wind up with an edge over the casino that’s close to the edge the casino has over the basic strategy players.
This means that you might be playing with an edge over the casino of 0.5% or 1%.
That’s not a huge edge.
But it’s big enough.
Projecting Hourly Win Rates, Loss Rates, and Bankroll Requirements
How much does the average blackjack player lose per hour?
The formula is simple:
It’s the number of bets per hour, multiplied by the average size of those bets, multiplied by the house edge.
At an average blackjack player, you might see 80 hands per hour. Say you’re betting an average of $5 per hand, and you’re not using basic strategy, you’re looking at $400 in action per hour and losing 5% of that–$20/hour.
A basic strategy player, on the other hand, might reduce the house edge of 0.5%. This reduces his hourly loss rate to just $2/hour.
A card counter might be operating with a 1% edge over the house though. At the stakes we’re talking about, he’s winning $4/hour.
But consider this:
He’s raising the size of his bets based on the count, so his average bet size won’t be $5. It’ll probably be closer to $20.
Now we’re looking at $16/hour.
That’s not a great living, by the way. You can probably make the same money working at an In N Out Burger, in fact.
Blackjack Probability Chart
But as your bankroll grows, so does the average size of your bets. The people making real money counting cards might be putting $100 per hand into action on average. That’s $8000 per hour, or $80/hour in winnings.
$80/hour is some real money.
But one thing card counters need to think about is something called “risk of ruin.”
Remember how I talked about short term variance, and how you can’t expect long term expectations to hold true in the short term?
Just because you have an edge when counting cards doesn’t mean you’re going to show winnings every hour. You’re going to have wild swings of luck. That 1% edge is an average over thousands of hands.
You need a big enough bankroll to handle those swings in fortune without going broke.
The bigger your bankroll, the less likely you are to go broke before your edge and the long term kicks in.
This probability is called “risk of ruin.”
Blackjack Statistics
The most conservative approach is to have 1000 units to bet with. If you’re averaging $100 bets, you need a bankroll of $100,000 to play. If you have that kind of bankroll, your risk of ruin is just 1%.
On the other hand, if your tolerance for risk is better than that, you could get away with a much smaller bankroll—maybe 200 units. You’d still need $20,000, but you’d be able to play at that level. Your risk of ruin goes way up, though—to 40%.
I suggest to Texas holdem players that they know they’re good enough to move up in level when they can increase their bankroll to the appropriate amount for that new level. I think this recommendation holds true for blackjack players, too.
If you have $2000, you should be playing for an average of $10/hand. If you’re succeeding at that level, you’ll eventually have a bankroll of $4000, and you can move up to $20/hand, and so on.
How conservative or aggressive you are is up to you and your temperament.
Conclusion
Blackjack probability is a fascinating subject with no end of subtopics you can discuss. I could just as easily have written about the probability of going bust with certain tables as I did with the approach I took. I just thought it would be more useful to tackle the subject of blackjack statistics from an aerial view.
Most people, frankly, aren’t cut out to be card counters. It sounds easier in theory than it is in practice. Blackjack in most casinos is fast-paced and confusing, especially if you’re new to the game. It’s hard to keep up with those numbers in your head without looking like you’re paying too much attention.
And don’t forget that part:
The casino is watching for card counters. To say they frown on counting cards is an understatement. Casinos will risk throwing out players they could profit from if they suspect them of counting cards.
In fact, I think most casinos would be better off if they lightened up on card counters. I know plenty of would-be card counters who make enough consistent mistakes that they only THINK they’re playing with an edge over the casino.
My guess is that the number of would-be card counters who are profitable in the long run make up between 5% and 10% of the total number of card counters in the business.
Probabilities In Blackjack
At any rate, knowing something about the probabilities behind the game makes it more fun, even if you have no interest in being an advantage player.
And if you’re not an advantage player, fun’s what it’s all about. Comparing the cost of that fun to the amount of enjoyment you get from playing is what smart recreational gambling is all about.
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