Poker Probability – Part 1: Basics of Probability

The market for uncertified physicians was not very big in 16th-century Italy. So when a young Girolamo Cardano found himself without a College membership or cash, he did what any reasonable person would do: he became a professional gambler.

Over his career, Cardano meticulously documented observations about the games he played. Specifically, he identified patterns that corresponded to wins and losses. Cardano’s observations – outlined in his work “Book on Games of Chance” – form the basis for what we now call mathematical probability.

Knowledge of probability allowed Cardano to earn a steady income from gambling, and professional gamblers today rely on probability to make winning bets. I doubt there’s one WSOP winner alive who lacks at least a basic understanding of probability.

Probability is the tool that allows us to win money at poker. So let’s take a look at how poker probability works, and learn how to apply it to our game. We also welcome you to check out our remaining 3 parts to this 4-part series on probability, which can be found below:

What Is Probability?

As a field of mathematics, probability is the study of uncertain events. What’s the chance that it will rain tomorrow? How likely am I to get an A on this test? If you’re dealt AA preflop, how often will you win the hand? The field of probability helps us to answer these questions.

When we obtain an answer to a question of chance, it’s called a probability. For example the probability of seeing rain tomorrow might be 50%. Half the time it will rain, and half the time it won’t.

In poker, probabilities help us to estimate the likelihood of certain events happening. For example the probability of getting aces in the hole is 1/221, or 0.45%. Knowing this particular probability helps us to guide our hand selection at the tables. More importantly, it helps us to guess our opponent’s hand distributions with stunning accuracy.

Representing Poker Probability

There are three main ways to express a probability:

• As a decimal (e.g. 0.03)
• As a percentage (e.g. 3%)
• As a fraction (e.g. 3/100)

All of the above expressions mean the exact same thing, they’re just written in different forms. How you choose to express probabilities boils down to personal preference.

On forum discussions and in poker strategy articles, you’ll most often see poker probabilities in percentage or fraction form. For example, the chance of As-2d winning against a random hand can be expressed as:

• 54.93% in percentage form
• 54.93/100 in fraction form

In mathematical calculations, we’ll most often use decimal form. It’s just easier to write that way, since calculators most often return decimal values. For example the expected value of going all-in against one opponent for $1 with As-2d is:

= $win* Pwin – $cost
= $2(0.5393) – $1
= $0.0786

Probabilities in Plain English

All these numbers can seem confusing at first. But if you start thinking about probabilities in plain English, you’ll eventually develop an intuition for what they are. That’s especially helpful with poker probabilities, where you often have to calculate them on the fly.

For example, say we have an event A with the probability P(A) = 1/52. This just means that for every 52 events, we can expect event A to happen exactly once. Said differently, event A will happen once per 52 cycles.

Numerator: # of times the event occurs
Denominator: # of iterations
Numerator / Denominator = Probability

Every probability has this same basic format, which means you can convert them easily to English descriptions. Try converting these example probabilities into plain-English descriptions yourself:

1. P(Dealt Aces) = 1 / 221
2. P(Aces Win Against Random Opponent) = 85.20 / 100
3. P(Kings Make Set on Flop) = 1 / 8.5

Check your answers at the bottom of the article.

How Does Probability Help With Poker Strategy?

Probability is useful in almost every aspect of poker strategy. To name just a few examples, probability can:

1. Tell us the likelihood of an opponent holding certain hands, given our reads and her actions.
2. Tell us how often we’ll be dealt strong vs. weak hands.
3. Tell us how often our strong hands or draws will win by showdown.

In the first case, we can transform our reads into mathematical evaluations of poker situations. This general process is called building a hand range, which is a probability distribution of an opponent’s possible holdings.

Once we calculate an opponent’s range, we can analyze the probability of our own holding’s winning against said range. And with that information, we can calculate the average earnings we’ll expect to make by betting, checking, calling, or folding. This allows us to play profitably 100% of the time, as long as we’re mentally sharp. Obviously this is a huge advantage when we’re trying to win at poker.

In the second case, we can figure out how long we’ll have to wait for good hands preflop. This information tells us how often we should expect to play with a card advantage against opponents. From this we can deduce how often we should be raising, folding, and calling preflop. Even better, we can deduce how often an opponent should be raising, calling, and folding; and then we can exploit our opponent’s strategy, depending on how weak or strong is action range actually is.

In the third case, we can calculate the profitability of playing an unmade hand. For example we might have one card to a flush on the turn facing an opponent’s bet. Probability can tell us how likely we are to make our flush. Using this probability, we can calculate our expected value: the exact dollar amount we’d expect to win by making the call. Once we’ve got this information, we’re absolutely certain whether or not a call is profitable in the long run!

Answers to Exercises

1. We’ll get aces once per 221 deals.
2. Our aces will win 82.20 times per 100 hands.
3. Our kings will flop a set once every 8.5 hands.

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