The Mathematics and Statistics of Blackjack

Blackjack is one of the most popular card games on the planet. This is likely due to how simple the game is to understand for beginners, which often leads to new players learning blackjack as their first card game. Regardless, while blackjack may seem like a simple game of chance on the surface, it’s actually a highly complex game that can involve a lot of math if you really look into it.

Advantage players, or gamblers/game players that recognize the statistics of the game, often use different mathematical techniques to determine how to play Blackjack successfully?


Luck may be the prominent factor in any casino game, but there is a way that you can “control” luck, known as variance to experienced players.  Variance is essentially the difference between the expected advantage of a game and the actual outcome.

For example, when you are card counting or playing the statistics of blackjack, there’s something known as expected value. This is what an experienced blackjack player would expect to make in a given time period. Variance then is the difference between what this person would expect to make and what they actually ended up with. Say they expected to make $2000 and they only ended up with $1750, the variance would be $250.

Variance is just one-factor statistical players keep in mind in figuring out how to play and how long to keep at it.

Standard Deviation (SD)

Standard deviation is a relatively common term in mathematics. Mathematically put, it’s the square root of variance. If variance is the difference, then the standard deviation is the measure of how large that difference is compared to the rest of the game’s factors.

For example, if you consistently experience a standard deviation of 2, then you know that you are either getting incredibly lucky or incredibly unlucky depending upon what direction the variance is (positive or negative). The close the SD is to 0, the more on-target your gae is playing out.

N-Zero (N0)

“No” is something you might not think about in Blackjack – it’s the theoretical number of hands that will be required before you get ahead in the game. This value is calculated using the standard deviation number you’re experiencing in a game. So, No is calculated using the formula Variance/(EV^2). Bringing the value of No into practicality, if you multiply your no value by 4, in a game where the standard deviation is 2, you’ll be left with a number telling you how many hours you’ll have to play to get ahead. That is, assuming you keep playing like you are.

Certainty Equivalence (CE)

Certainty Equivalence is otherwise known as the risk-adjusted return of a game. It’s calculated by taking the expected win rate of a game and altering it based on the level of risk you’re willing to take on. Essentially, this value will tell you if it’s worth it to take on a current game.

The basic equation of figuring CE. EV-( (bet size*Standard Deviation)^2)/(2*Kelly factor*Bank Roll)

Think about it like this, would you take $100 in cash or take a 50/50 shot at $200. CE helps you evaluate that question with math, not emotion.

As you’re likely realizing, there’s actually an impressive amount of math that can be applied to blackjack and other card games. While counting cards and applying statistics to games may not be your thing, there are other ways at increasing your odds at winning blackjack. Take a look at this article here.

Trevor is a civil engineer (B.S.) by trade and an accomplished writer with a passion for inspiring everyone with new and exciting technologies. He is also a published children’s book author and the producer for the YouTube channel Concerning Reality.

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