1. Chances Of Drawing A Blackjack Free
2. Chances Of Drawing A Blackjack Machine
3. Chances Of Drawing A Blackjack Online
4. Chances Of Drawing A Blackjack Game

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.

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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.

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.

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. Get money right now. 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.

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:

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.

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.

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.

Chances Of Drawing A Blackjack Free

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.

Chances Of Drawing A Blackjack Machine

Effects of Different Rules on the House Edge

Chances Of Drawing A Blackjack Online

The conditions under which you play blackjack affect the
house edge. For example, the more decks in play, the higher the
house edge. If the dealer hits a soft 17 instead of standing,
the house edge goes up. Getting paid 6 to 5 instead of 3 to 2
for a blackjack also increases the house edge.

Luckily, we know the effect each of these changes has on the
house edge. Using this information, we can make educated
decisions about which games to play and which games to avoid.

Here’s a table with some of the effects of various rule
conditions.

These are just some examples. There are multiple rules
variations you can find, some of which are so dramatic that the
game gets a different name entirely. Examples include Spanish 21
and Double Exposure.

The composition of the deck affects the house edge, too. We
touched on this earlier when discussing how card counting works.
But we can go into more detail here.

Every card that is removed from the deck moves the house edge
up or down on the subsequent hands. This might not make sense
initially, but think about it. If you removed all the aces from
the deck, it would be impossible to get a 3 to 2 payout on a
blackjack. That would increase the house edge significantly,
wouldn’t it?

Here’s the effect on the house edge when you remove a card of
a certain rank from the deck.

These percentages are based on a single deck. If you’re
playing in a game with multiple decks, the effect of the removal
of each card is diluted by the number of decks in play.

Chances Of Drawing A Blackjack Game

Looking at these numbers is telling, especially when you
compare these percentages with the values given to the cards
when counting. The low cards (2-6) have the most dramatic effect
on the house edge. That’s why almost all counting systems assign
a value to each of them. The middle cards (7-9) have a much
smaller effect. Then the high cards, aces and tens, also have a
large effect.

The most important cards are the aces and the fives. Each of
those cards is worth over 0.5% to the house edge. That’s why the
simplest card counting system, the ace-five count, only tracks
those two ranks. They’re that powerful.

You can also look at the probability that a dealer will bust
based on her up card. This provides some insight into how basic
strategy decisions work.

Perceptive readers will notice a big jump in the probability
of a dealer busting between the numbers six and seven. They’ll
also notice a similar division on most basic strategy charts.
Players generally stand more often when the dealer has a six or
lower showing. That’s because the dealer has a significantly
greater chance of going bust.

Summary and Further Reading

Odds and probability in blackjack is a subject with endless
ramifications. The most important concepts to understand are how
to calculate probability, how to understand expected value, and
how to quantify the house edge. Understanding the underlying
probabilities in the game makes learning basic strategy and card
counting techniques easier.

Blackjack strategy, applied perfectly, makes blackjack one of the most potentially lucrative games in the casino. If you play blackjack by the numbers, the House only has an edge of half a percent. That means if you spend $100 at a blackjack table in an hour, and you play using perfect basic strategy, you only stand to lose about 50 cents. The low house edge makes blackjack a truly cheap form of entertainment.

Blackjack rules give the game a certain purity of probability, a statistical beauty that statisticians and math nerds have exploited over the years to come up with blackjack strategy. Odds and probabilities are a huge part of winning at blackjack, so if you want to win big money at the casino, you have to understand the game’s odds and probabilities.

Blackjack Math

Blackjack strategy cards are sold at every casino gift shop in Vegas. These cards range from simple “if ___ happens, do ___” suggestions all the way up to cards with multiple rule change options on strategy. Casinos sell these cards because most people are more interested in playing a game and making big gambles than methodically picking apart the House using blackjack math and a little plastic card.

Blackjack math gives players an informed opinion about what move to make next. Rather than taking a random gamble, blackjack strategy uses odds and probabilities to tell you the move most likely to end in you beating the House. Blackjack strategy tips are more than educated guesses, they have hard math to back them up.

Here are some blackjack math basics: any deck of 52 cards contains a certain number of cards worth 10 points and a certain number of cards worth their face value. The odds of pulling a single face value card, like a 7 or a 9, is 4 out of 52. or 7.69%.

There are 16 cards in a standard deck with a value of 10 points: four 10s, four Jacks, four Queens, and four Kings. That means the odds of drawing a 10-point card are 16 out of 52, or 30.8%.

Players are four times more likely to draw a card worth ten than any other card value. 10 point cards outnumber all others, so most blackjack strategy is built around figuring out the probability of the player or the dealer drawing those high point value cards.

Using Blackjack Odds & Probabilities

The math behind the game of blackjack has a direct effect on proper blackjack strategy. We just saw how much more likely a blackjack player is to draw a 10 point card than any other card, but how does that effect blackjack strategy? At the most basic level, knowing that you’re more likely to draw a card worth ten points makes a point total of 12 or more an anxious game decision.

Another example of how blackjack odds play a big part in player strategy is what blackjack experts call bust probability. Bust probability is a number that indicates how often a certain point total will lead to a bust. Since the blackjack dealer doesn’t have free will, and must stand at 17 in most casinos, his bust likelihood is higher when his point total is 16, because dealers have to hit at 16. This rule also affects early dealer point totals—imagine a dealer with an upcard of 6. He has to hit, and we already know how much more likely you are to draw a ten point card than any other, putting him in great danger of drawing a 16 and being forced to hit into a bust.

Other Blackjack Odds & Probabilities

Blackjack math is straightforward, since each card in the deck has its own consistent point value and there’s a clear line drawn between winner and loser. Applying a little bit of math, blackjack experts have figured out the odds of being dealt different types of blackjack hands. Your chance of drawing a natural 21, and winning instantly, is 4.8%. Play 100 hands of blackjack and you’re likely to only be dealt a natural blackjack 5 times.

That makes blackjack sound like a hard game to beat, but the truth couldn’t be more different. Those same odds show that the most common hand dealt in blackjack is the decision hand, any point total between 1 and 16. These hands are dealt 38.7% of the time, meaning that out of every 100 hands you play, you’ll have to make a decision only about 40 times. In the other 60 hands, the decision will be made for you.

If you play with basic strategy, and make the right play according to that strategy for each of those decision hands, you can minimize the house edge.

A final trick to using blackjack odds to beat the House is considering two crucial statistics at once. We’ve already talked about dealer bust probability and player edge against the House. Just combine the two statistics for an idea of when you have the best chance to beat the blackjack dealer.

For each dealer upcard, mathematicians have figured out their likelihood of drawing to a bust. For example, if a dealer’s upcard is a 2, his bust probability is 38%. Factor in the player’s advantage gained by using basic strategy against each upcard, and you get a good look at when you have your best shot at beating the House.

The House is in the most danger of busting or losing to basic strategy when the dealer’s upcard is a 5 or 6. When the dealer shows a 5 upcard, his bust probability is 42.89%. Since players have a 23.2% advantage against that upcard if they play with perfect strategy, it’s clear that when a dealer shows a 5, players are more likely to beat the dealer.

Using blackjack math and odds to your advantage against the casino is called basic blackjack strategy. Make sure and pick up a blackjack strategy chart next time you hit the casino, and work on memorizing the proper play for different game situations. Following the lead of smarter blackjack players who have gone before you is important if you like making money on the gambling floor.