Odds & Probability in Blackjack
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 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. 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.
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.
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.
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
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
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 & 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.