Everyone knows that the house edge in blackjack is one of the lowest in the casino. It varies based on what kinds of rules are in effect at a specific property. In Las Vegas, you can find games with a house edge as low as 0.26%, although closer to 1% is more common.
But a number of blackjack odds and probabilities are important to understanding the game.
This post starts by explaining some of the basics of probability in general. It goes on to explain how the odds and probabilities work specific to blackjack.
What Is Probability and What Are Odds?
Probability is the branch of math that studies the likelihood that an event will or won’t happen. Probability is also that likelihood expressed numerically.
The probability that something will happen is always a number between 0 and 1. If something can never happen, it has a probability of 0. If something will always happen, it has a probability of 1.
You can express this number in multiple ways. One of the most common is as a percentage. Something with a probability of 100% is the same thing as something with a probability of 1.
Another way to express a probability is as a fraction. If something will happen 25% of the time, you can also say that it’s probability is 1/4.
Odds express the number of ways something can’t happen with the number of ways it can. For example, something with a probability of 1/4 can be expressed in odds format as being 3 to 1.
Some Specific Examples Related to Blackjack
Since all the cards at the blackjack table have a value of between 1 and 11, you can say that there’s a 100% chance that the next card will be worth between 1 and 11 points.
There’s also a 0% chance that the next card will be worth 12 points. It’s impossible.
But let’s talk about a more practical example:
You have a 10 and a 6 for a hard total of 16.
What is the probability that you’ll bust that total if you take a hit?
Assuming the dealer has a 6 showing, we know 3 cards, and there are 48 cards left in the deck.
Any card ranked 6 or higher will bust the hand.
How many of those are there?
There are 2 6s. There are also 4 7s, 8s, and 9s. And there are 15 10s left in the deck. That’s a total of 29 cards that will bust this hand.
The decks also has 4 5s, 4s, 2s, 2s, and aces in it. That’s 20 more cards.
The probability of busting in this situation is 29/49, which amounts to 59.18%.
The probability of not busting in this situation is 40.82%.
Add the 2 together, and you get 100%. That makes sense, because of course you’re either going to bust or not bust. One of the 2 must happen if you take an additional card.
How Probability Works with Basic Strategy
Luckily, you don’t have to do calculations like that last example on the fly. Computer programmers and mathematicians have already put together these numbers to create “basic strategy”.
Put simply, basic strategy is just the knowledge of the best possible decision in every possible situation in the game of blackjack. The low house edge that everyone quotes for blackjack assumes that you’re using basic strategy.
If you ignore basic strategy and just do your own thing, your results will likely be more along the lines of 4% or 5%.
The Effect of the House Edge in the Game of Blackjack
The house edge is a long-term prediction of how much money you’ll lose on average per bet that you make. If you say that the house edge is 1%, this means that over enough hands, you’ll lose an average of $1 for every $100 that you bet.
The house edge is a long-term expectation only. In the short run, it’s not only unlikely to hold true, it’s impossible to hold true.
You simply can’t average a loss of $1 on a hand or 2 of blackjack if you’re betting $100 per hand.
On a single hand of blackjack, you can lose $50 if you surrender. You can lose $100 if you bust or if the dealer beats your total. You can win $100 if you beat the dealer or if she busts. You can win $150 if you get a blackjack. If there’s a push, you can win $0 or lose $0, however you want to look at it.
None of those results come close to the mathematical expectation, though.
But over thousands of hands, your results should start to resemble the mathematical expectation. You would average your net loss or net win by the total number of hands you played to get your average win or loss per hand.
Casinos use these numbers to predict how much you’re worth to the casino over time.
Let’s say you’re playing 70 hands of blackjack per hour at $100 per hand in a game where the house edge is 1%. You’re expected to lose $1 per hand, so the casino expects you to lose $70 per hour in the long run.
In the short run, over a few hours, or even a few dozen hours, you might win $50 per hour or lose $100 per hour. But the more hands you get in, the closer those results get to the mathematical expectation.
Thinking about Blackjack Probabilities in a Variety of Situations
Here’s one easy probability to calculate, and we figure it out for one possible total already (16).
What’s the probability that you will bust your hand if you take a hit?
If you have a total of 21, there’s a 100% chance of busting.
But if you have a total of 20, the probability of busting drops to 92%.
There’s an 85% probability of busting a total of 19.
The probability of busting an 18 is similarly high–77%.
You would never hit one of these hands, in fact. The probability of busting is too high. It doesn’t matter what the dealer’s face-up card is.
But once you get to a low enough total, the probability of busting gets very small indeed. If you have a total of 11 or less, for example, the probability of busting from your next card is 0%.
If you have a total of 12, the probability of busting is 31%.
In fact, for every point difference in the total of your hand, the probability of busting goes up by 7-8%. It starts at 31% and goes up to 100%.
Another Interesting Set of Blackjack Odds to Look At
You might be interested to know that about 35% of the time, you won’t have to make a decision in blackjack. That’s how often you’ll have a hard total of 17 or higher. You never hit those hands.
Removing Cards from the Deck
One of my favorite blackjack statistics relates to the change in the house edge when a card is removed from the deck. If you’ve read any of my posts about counting cards, you’ll know that the odds tilt in your favor as the low cards get dealt. They tilt back in the casino’s favor as the high cards get dealt.
It turns out that the 5s have the biggest effect on the house edge. Every 5 you remove from a single deck of cards reduces the house edge by 0.67%.
The implications of that are pretty interesting. Suppose you’re playing in a game where you face a 1% house edge. If you took all the 5s out of the deck, you’d have an edge over the casino of over 2%.
The first book about blackjack I ever read was The World’s Greatest Blackjack Book by Lance Humble. My favorite anecdote in the book was about how he was playing in an underground casino with Lawrence Revere (another famous blackjack author).
Apparently Humble and Revere were talking in the bathroom, and Revere was bragging that they were going to get the better of the casino. Then he showed Humble that he’d removed the 5s from the deck.
The 4s and the 6s have similar effects on the house edge–0.52% and 0.45% respectively.
2s and 3s each add 0.4% and 0.43% to the house edge when they come out of the deck.
It’s no coincidence that almost all counting systems count any card of 2, 3, 4, 5, or 6 as at least a +1.
But on the other side of things, the aces and 10s matter a lot, too. Every time you remove an ace from the deck, the house edge goes up by a whopping 0.59%.
The 10s are almost as important. Removing a 10 from the deck increases the house edge by 0.51%.
The 7s, 8s, and 9s have an effect, too, but it’s negligible compared to the other cards. In fact, removing an 8 from the deck only affects the house edge by 0.01%, which is negligible.
Removing a 9 from the deck is good for the casino, and removing a 7 from the deck is good for the player. But the effect is only 0.15% and 0.3%, respectively.
Most card counting systems ignore the 7s, 8s, and 9s for this reason. In fact, the numbers are close enough that simply adding and subtracting 1 is enough to get an edge over the casino. Just raise your bets when you have the edge, and you can beat the casino, too.
You can find card counting systems which try to increase the accuracy of their results even more. I think this is a mistake. Over-complicating something as simple as card counting is counter-productive.
What the Dealer’s Up Card Does to Your Advantage or Lack Thereof
One of the things to think about is how likely it is that the dealer will bust. If you’ve studied a basic strategy chart and thought about it, you might have already realized this:
The dealer is more likely to bust when he was a 6 or lower as his up card.
In fact, the 2 worst cards for the dealer are the 5 and the 6. He has a roughly 42% chance of busting with either of those up cards.
That percentage goes down by about 2% for every point drop. An up card of 4 gives the dealer a roughly 40% chance of busting. With a 3, the chance of busting drops to a little less than 38%. And with a 2, the probability of busting drops to 35%.
The dealer’s probability of busting with a 7 or higher is much smaller. It’s 25% with a 7. With an 8 or 9, it’s around 23%. With a 10, the dealer has a probability of busting of about 21%.
But if the dealer has an ace as his up card, the probability of busting is even lower–between 11% and 12%.
Understanding this helps you understand why basic strategy works the way it does. A lot of times when you stand on a total that seems too low, you’re doing so because the dealer’s probability of busting is low.
What Is the Player’s Advantage or Disadvantage Based on the Dealer’s Up Card?
We often talk about the 1% house edge that the casino has overall. But did you know that the player has an advantage over the dealer in a lot of situations?
In fact, the player has the edge over the casino if the dealer has any card 8 or lower showing. And that advantage is significant, too. If the dealer has a 5 or 6 showing, the player’s edge over the casino is 23%–if the player is following basic strategy.
Unfortunately for the player, the casino has a huge edge when the dealer has a 9, 10, or ace showing. In fact, if the dealer has a 10 or an ace showing, the casino’s edge over the player is 16%.
And there are more 10s in the deck than any other individual card.
Of course, these numbers only apply if you’re actually following the basic strategy. Every deviation you make from perfect basic strategy puts the numbers back in the casino’s favor.
Blackjack odds and probability are endlessly fascinating. For most players, understanding that the house edge is less than 1% is enough to convince them to play blackjack instead of other games. It’s also enough to make sure they use basic strategy every time they play.
When you start to get a grip on what some of these numbers really mean, you start to become more intuitive about the game and how you should play it.
Counting cards becomes less theoretical and more practical as you understand the odds better, too.
And for the most part, probability is mostly just a matter of division, fractions, and percentages. If you were good at those subjects in school, you’ll probably also be good at blackjack odds and probability in real life, too.