Blackjack Odds and Probability

Author: GamblingSites.org | Last Updated: April 2023

Blackjack is one of the most popular casino games in the world, both online and at in-person casinos. The game has seen massive success because of it’s easy-to-learn ruleset and large payback potential. But, what most people don’t realize is that it’s also one of the best games to play because of it’s low house edge and friendly probability for success.

Blackjack runs on math, even though it might not be visible to the untrained eye. The game is all about probability. How likely is it that you’ll hit a card that gives you 21? How likely is it that the dealer reveals their card and busts? What are the odds on winning multiple hands in a row? All of these questions are rooted in the firm probabilities of the game, and intermediate players should start to understand how they work to get better at blackjack.

Here we’ll give a basic run down of blackjack odds, as well as the probability of certain outcomes in blackjack. We’ll also prep you for some tricky situations where the odds might be against you, but you’re still able to come out on top. The odds of winning blackjack aren’t always in your favor, but with basic strategy and a firm grasp of the logic behind the game, you can move to more advanced tables and win even more real money with blackjack.

An Introduction to Blackjack Odds & 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.

Games like slots are largely driven by random numbers, so it’s difficult to use probability to determine the event in which you win. But, games like blackjack, poker, and baccarat have a finite number of card combinations, making it easier for mathematicians to lay out a set of odds and probabilities.

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.

Alternatively, 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.

You can express probability as a fraction or as 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.

Alternatively, 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.

Blackjack 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. Let’s take the game detailed above as an example.
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.

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.

Odds of Winning Blackjack with Unfavorable Hands

There are a couple of hands in blackjack that are particularly difficult for players to win. These hands just happen to be the right combination of cards at the right time, and inexperienced players will often freeze or panic when they come across these hands. Let’s take a look at some of these hands through the lens of probability and determine what basic blackjack strategy suggests we should do.

Hard 16 – Hard 16 is risky business, but if the dealer has a 7 or higher as the up card, you should hit. The dealer might have a 9 or a 10, and in this case, you’re better off surrendering. The dealer has around a 21-23% chance of busting, so it’s easy to justify a surrender.

Player 12 vs Dealer 4 – Having a 12 hand is tough because you can hit a 10/face and bust, or a 9 and get 21. It’s right on the threshold. And when the dealer puts up a 4, it can make your choice a lot more difficult. But, a 4 means the dealer has a 40% chance of a bust, so standing will hopefully result in a win for you. However, you can risk a hit, because your percentage of busting with a 12 is less than the dealer’s.

Player 15 vs Dealer 10 – Honestly, you have about the same chance of winning when you hit as when you stand. It’s always good to assume that the dealer will reveal a winning hand, and with a 10, they only have a 21% chance of busting. You’re better off hitting here and crossing your fingers than standing and risk being beaten out. They only need to draw a 6 to win.

Player 12 vs. Dealer 3 – While you might think this hand is similar to the 12 vs 4 scenario above, it’s not as worrisome, but still risky. However, your chance of busting is less than the dealer’s, so it’s advisable that you hit at least once.

The Blackjack House Edge

If you’ve been playing casino games for a while, you’ve probably heard about the house edge. But what is the house edge in blackjack?

In simple terms, the house edge is the advantage the casino has over the player. 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.

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.

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?

Just like how the probability changes from 1/52 to 51 to 1, the percentage of house edge changes based on the cards still in the deck. 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.

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.

The Probabilities of Busting in Blackjack

One of the most common uses of probabilities is for determining when a player or dealer will bust in blackjack. Truthfully, if you’ve played the game for long enough, you’ll get a clear sense of what card combinations will come out with a bust, and which ones won’t. But, it can be helpful to visualize exactly how the probability changes based on the cards in play.

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.

Blackjack Odds FAQs

Summary and Further Reading

Once you learn blackjack odds and probability, you’ve opened up a door to a future of more profitable gambling. 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. Check out some of our other strategy pages if you’re looking to continue improving your blackjack skills.