The most common variations of card counting in blackjack are based on statistical evidence that high cards (especially aces and 10s, and to a lesser extent 9s) benefit the player more than the dealer, while the low cards, (3s, 4s, 6s, and especially 5s, and to a lesser extent 2s and 7s) help the dealer while hurting the player. 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.

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:

Roulette Odds Wikipedia

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”.

• 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.

Poker Odds Wikipedia

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.

Arts + Entertainment

You too can win a fortune counting cards. Probably.

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For most people, gambling is a fast way to lose money while slurping down free (well, sort of free) drinks. But for a few years in the 1980s, Bill Kaplan and a few associates figured out a way to clean house and rake in cash, and thus, the legend of the MIT Blackjack Team was born. Helmed by Kaplan (who didn’t actually go to MIT), the MIT team used card counting and a strict system to carve out an advantage against the dealer that they ruthlessly exploited. With the Encore Boston Harbor Casino throwing open its doors, I wondered: Do you have to be an MIT-level genius to pull off a scheme like this, or can any schlub with a will to learn and dollar signs in his eyes make do? Kaplan was kind enough to hop on the phone and break down just what it takes to beat the house.

(The interview has been lightly edited for length and clarity.)

You founded the MIT Blackjack Team, which became famous for beating casinos and winning huge amounts of money. How did that all start?

When I first got out of college—I went to Harvard—I was supposed to go right into Harvard Business School. Instead, I deferred for a year, moved out to Vegas, and started a blackjack team. The thing about blackjack is that you can actually win at this game. So, I ran a team for a year, and kept running it while I was in business school. Eventually though, everyone was burned out, all the casinos knew our faces. And at some point, I was sitting in a Chinese restaurant in Cambridge talking about all this and some MIT kids came over said, “Are you talking about blackjack? We’ve been playing for a few months, trying to make money.”

Okay, but how did you get into blackjack at all?

I read this book by Edward O. Thorp called Beat the Dealer. He said if you crunched all the numbers, the game actually could be beaten, if you kept to a certain system. My background was in stats, applied math, and computer science, and I was curious: How can you win at a game that casinos win billions of dollars at?

So how do you do it?

Counting cards lets you see whether the player or the casino has the odds, and then you use a betting strategy, how much you can bet on each hand. Card counting tells you how to optimally play each hand, based on what cards are remaining in the shoe and your expected advantage. Then, there’s a whole money management piece of it.

Was there something that set the MIT team apart?

Most teams fail on the money management. They never make it to the long run.

Let’s say we’re flipping a quarter with a 51 percent chance of landing on one side. If you’re betting a dollar on each flip, but you only have five dollars to your name, you’re going to get wiped out. So, there’s a whole piece of how much you should be betting so you can not get wiped out. If I play 10,000 hands, it’ll show me the advantage.

Is this something that you need to be an MIT genius to figure out, or do you think me and the BoMag staff can put together a team?

Honestly, just about anyone with a high school education can learn the game. It just happened to be, everyone on the team was from Harvard, MIT, Princeton, University of Chicago. You could teach someone—there eventually were a couple of people who played on the team who just had a high school education. It’s not that difficult. Doing all the strategy and money managing can be pretty tough, but as a player, it’s just some memorization.

What if I tried to give it a go on my own?

Blackjack Odds Winning

It’s hard to do, as an individual. One of the reasons you play as a team, you can pool all your capital, but you’re able to get to the long run sooner. As a player, your advantage is about one percent. If you’ve got 30 people playing and putting in eight hours, playing in accordance with the guidelines, you get to the long run that much faster.

Did you ever get kicked out of a casino?

At every casino, eventually you get kicked out. By the time we were playing, most were public companies. If you saw [casino staff] coming, they’d say, “Go play somewhere else, you can’t play blackjack here anymore.” There were some people, not on our team, who got roughed up. Casinos did a bunch of things that were illegal back then. They’d steal our money out of safety deposit boxes, for instance, and we’d have to hire lawyers to get it back.

Did you ever try to beat another game?

Blackjack is the only game that you can legitimately beat. If you play basic strategy, you only have a half percent disadvantage. If you count the cards, the advantage switches to the player, even though the odds are switching and moving as the cards are being played out. It’s the only one you can really win.

What made the MIT team so good?

There are tens of thousands of people who tried to win at the game, but MIT was the only team who really won year over year, because we ran it like a business. Training, extensive training, checkout procedures, two hours of perfect play, leaving the table right. It was really run more tightly than most businesses.

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