Poker Probability

Introduction

This article provides an overview of poker probability, explaining the concepts and how probability applies to poker.  It describes the probabilities for various poker hands, including one pair, full house, and more, across different poker variations. Additionally, it introduces poker probability calculators that can help players evaluate their winning chances based on their hole cards and community cards. It also provides guidance on understanding how to calculate and apply poker probabilities, which can help in making strategic decisions while playing the game.

What is Probability?

In layman’s terms, probability means “possibility of happening”. Generally used in mathematics, probability explains how likely or to an extent something will happen. 

Understanding the possible outcomes becomes crucial as we discuss the probability of something happening. For instance, if you toss a coin, there are two possible outcomes: you’ll get a head or a tail. Here, the probability of getting a head or a tail is 50-50 each (one outcome out of the two). However, if you throw a dice, there are six possible outcomes, i.e. you can get 1,2,3,4,5 or 6. In this case, the probability of getting one of them is 1/6.

Poker Probability

In poker, the probability is the likelihood of a particular event happening. The most common events in poker are:

• Dealing a certain hand
• Winning a hand
• Making a specific hand (e.g. a flush or straight)
• Dealing a hole card
• Flopping a Poker Hand

To calculate probabilities in poker, you need to know two things:

• The number of cards in the deck
• The number of cards that will help you achieve your desired outcome

For example, if you want to calculate the probability of being dealt a pocket pair (two cards of the same rank), you would do the following:

• There are 52 cards in the deck.
• There are 13 different ranks, each with 4 cards, one for each suit.
• Therefore, to complete a pocket pair, the first card can be any among the 52 cards and the second card can be any of the three remaining cards of the same rank, i.e., 156 combinations. As K♦K♥ is the same as a K♥K♦, a total of 78 pocket pairs are possible.

Therefore, the probability of getting a pocket pair is 78/1326, or approximately 5.9%.

Poker Hands Probability

Poker Probability Calculator

You can use a poker hand probability calculator to more easily calculate your poker hand win probability. These calculators consider the cards you hold, the community cards, and the number of opponents you are playing against to give you a precise estimate of your chances of winning the hand.

Probability of One Pair in Poker

One of the most common poker hands is one pair with two cards of the same rank. The probability of getting one pair in a five-card poker hand is approximately 42.3%. However, the probability of getting one pair in a three-card poker hand is much higher, at 16.9%.

3 Card Poker Probability

In 3-card poker, players receive 3 cards and intend to have the highest-ranked hand possible. The probability of getting a straight flush, the highest-ranking hand in 3-card poker, is approximately 0.22%. However, the probability of getting a pair is much higher, at approximately 16.9%.

4 Card Poker Probability

4-card poker is similar to 3-card poker, but players are dealt 4 cards instead of 3. The probability of getting a straight flush in 4-card poker is approximately 0.02%, and the probability of getting a pair is approximately 28%.

5 Card Poker Probability

In 5-card poker, players receive 5 cards, and the aim is to have the highest-ranked hand possible. The probability of getting a royal flush, the highest-ranking hand in 5-card poker, is approximately 0.0032%. However, the probability of getting one pair is much higher, at approximately 42.3%.

Full House Poker Probability

A full house is a hand made of 3 cards of one rank and the other 2 cards of another rank. The probability of getting a full house in a 5-card poker hand is approximately 2.6%.

Poker Hand Probability Texas Holdem

Texas Holdem is one of the most popular variants of poker. The probability of winning a hand in this game depends on the strength of your hand, the number of opponents, and the community cards. The probability of getting a pair in Texas Holdem is approximately 42.3%, and the probability of getting a full house is approximately 2.6%.

How to Calculate Poker Probability

To calculate poker probability, you consider your hole cards, the community cards on the table, and the number of opponents you are playing against. You can then use mathematical formulas to determine your probability of winning the hand.

Let’s start by understanding the odds of being dealt with certain starting hands in poker and the odds of flopping poker hands. These can be calculated for most scenarios and are a very effective tool while playing.

Odds of Being Dealt Specific Hole Cards in Poker

In poker, particularly Texas Hold'em, the odds of being dealt certain hole cards or the initial two cards are critical for developing strategy in the pre-flop stage. Understanding these probabilities and odds helps make informed decisions about whether to bet, raise, fold, or call during the pre-flop.

So, while dealing with the hole cards, the first card out of the deck can be either of the 52 cards, while the second hole card can be either of the remaining 51 cards. But since, for example, a K♦Q♣ is the same as Q♣K♦, the total number of combinations will be calculated as (52*51/2); 1326 unique combinations are possible.

Now that we know the total number of combinations possible, let’s examine some specific cases to understand this better.

Pocket Pairs:

These are pairs of the same rank, such as a pair of Kings or 3s. Since there are 52 cards in a deck, and you are getting two of them as hole cards, the odds of getting a specific pocket pair (like two Aces) is 1 in 221, or 0.45%. To calculate this, consider that the first card doesn't matter (any of the 52 cards will do), but the second card must be one of the three remaining cards of the same rank out of the 51 unseen cards. Therefore, for any specific pair, the probability of getting a pocket pair is (4/52) * (3/51); 1/221.

Suited Cards:

Getting two cards of the same suit improves the potential of making a flush. The odds of getting two cards of the same suit are 23.5%. It is because 12 cards remain from the same suit after receiving the first card out of the remaining 51 cards, making 52*12 suited combinations. Nevertheless, a K♦Q♦ is the same as a Q♦K♦. Thus, the total number of suited combinations becomes (52*12/2)= 312. Hence the probability (312/1326); 4/17.

Unsuited and Unpaired Cards:

Cards of different suits are quite common. The probability of getting two unsuited cards (excluding pairs) is 12/17 or 70.6%. It is determined by considering that 36 (excluding 3 cards of the same rank in the remaining 39 unsuited cards) of the remaining 51 cards are of a different suit and rank than the first card, making 52*36/2; 936 suited combinations. Hence, the probability (936/1326) is 12/17.

AK Suited:

An example of a strong starting hand is being dealt an Ace and a King of the same suit. The odds of this specific hand are 1 in 331, or 0.3%. It is calculated knowing there are 4 ways to get an Ace and King of the same suit out of the 1,326 possible starting hands. Hence, the probability (4/1326) is 1/331.

AK Unsuited:

Let’s look into a case of a starting hand being dealt an Ace and a King of the different suits. The odds of this specific hand are 1 in 110.5, or 0.9%. This is calculated knowing there are 12 ways to get an Ace and King of the different suits out of the 1,326 possible starting hands. Hence, the probability (12/1326) is 2/223.

Pocket Aces:

Again, one of the strongest hole cards could be a pair of Aces. The odds of getting a pair of Aces are 1 in 221 or 0.5%.  This is calculated knowing there are 6 ways to get two Aces of the different suits out of the 1,326 possible starting hands. Hence, the probability (6/1326) is 1/221.

QQ+(Pocket Queens and above):

In a deck, any pocket pair has a total of 6 combinations. For a pair of Qs, possible combinations could be Q♠Q♣, Q♦Q♣, Q♥Q♣, Q♠Q♥, Q♠Q♦, Q♦Q♥. Similarly, if you want to calculate the odds of the hole cards being QQ+, there will be pocket pairs of Qs, Ks and As, hence 18 combinations. So the probability (18/1326) is 3/221.

The Odds of Flopping a Made Hand:

Focusing on the odds of flopping a made hand in Texas Holdem, it is essential to consider the hands you might aim for and their respective probabilities.

To make a flop, you have to draw 3 cards from the remaining 50. Therefore, you can calculate the total number of possible flops by counting or applying combination concepts. There are 19,600 ways in which all the possible flops can be made.

Let's examine a few scenarios, including examples of hole cards, the resulting flop, probabilities, and odds for each situation.

Flopping a Pair

Example:

Hole Cards: 7♦ K♠

Flop: K♦ 4♠ 9♣

Probability: 30.99%

Odds: About 2.08 to 1

You have around a 1 in 3 chance of making a pair on the flop when you hold two unpaired cards. There are 6072 different ways to flop a pair in the case of Texas Hold’em. It is one of the most common ways to make a hand in Texas Holdem. So, the probability of flopping a pair is 6072 out of 19,600 ways, i.e., 30.99%.

Flopping a Set (Three of a Kind)

Hole Cards: J♣ J♠

Flop: J♦ 6♠ 2♠

Probability: 11.51%

Odds: About 7.69 to 1 

The odds of making a set on the flop when holding a pocket pair are roughly 1 in 8. There are 2,256 different ways to flop a Three-of-a-Kind in the case of Texas Hold’em. So the probability of flopping a set is 2,256 out of 19,600, i.e., 11.51%. Having a pocket pair and flopping a third one to make a set is a strong position, but it happens less frequently.

 Flopping Two Pair

Hole Cards: Q♠ 8♣

Flop: Q♦ 8♦ 4♥

Probability: ~2% 

Odds: About 49 to 1 

The chance of flopping two pairs when holding two unpaired cards is relatively rare, reflecting its strength as a hand in many situations. There are 396 different ways to flop a Two Pair in the case of Texas Hold’em. So the probability of flopping a two pair is 396 (3 suits of first hole card * 3 suits of second hole card * 44 remaining cards) out of 19,600, i.e., 2.02%. Having two unsuited cards and flopping a two-pair to make a set is a strong position, but it happens only once out of 49 occurrences.

Flopping a Straight

Hole Cards: 10♠ J♦

Flop: 9♣ Q♠ K♦

Probability: ~1.3%

Odds: About 77 to 1

Flopping straight (5 cards in a sequence) requires a specific setup of hole cards and the flop, making it quite rare, around 1.3%. The probability varies significantly based on the structure of your hole cards, e.g., connector cards (8-9) or gap cards (7-9).

Flopping a Flush

Hole Cards: A♠ K♠

Flop: 2♠ 6♠ 9♠

Probability: ~0.84% 

Odds: About 118 to 1

Flopping a flush requires you to have two suited cards in your hand and three more of the same suit to appear on the flop. There are 165 different ways to flop a Flush in the case of Texas Hold’em. So the probability of flopping a flush is 165 out of 19,600 total ways of making a flop, i.e., 0.84%

Flopping a Full House

Hole Cards: Q♥ Q♦

Flop: Q♠ 7♣ 7♦

Probability: ~0.14% 

Odds: About 693 to 1

Flopping a full house in poker is extremely rare. This scenario generally occurs when you start with a pocket pair (Q♥ Q♦), and the flop (Q♠ 7♣ 7♦) brings another pair (7♣ 7♦) plus a third card (Q♠) to match your own.

Another scenario could be when you start with unpaired hole cards (Q♠ 7♣). And in the flop (Q♥ Q♦7♦), one hole card meets another card of the same rank (7♦) to complete a pair (7♣7♦) while the other completes Three-of-a-Kind (Q♥ Q♦) with the presence of 2 cards of the same rank.

Flopping Four of a Kind

Hole Cards: 5♣ 5♠

Flop: 5♥ 5♦ 9♣

Probability: ~0.24% 

Odds: About 407 to 1

It is one of the rarest flops in poker, where you hold a pocket pair, and the flop brings the other two cards of the same rank.

When you start with a pocket pair in the hole cards, you need the remaining two cards of the same rank to make the four-of-a-kind in the flop. The third card in the flop can be any of the 48 cards left. Hence, the probability of flopping a Four-of-a-Kind is 0.24% (48/19600).

Remember, these probabilities and odds are specific to the flop and do not consider the turn and river. Poker consists of skill, strategy, and luck, and understanding these odds can significantly enhance your decision-making process.

Working Out the Probability of Draws

In poker, ‘out’ refers to the card that can improve your hand strength. For example, for a player who is hoping for a gutshot (5th card to complete a straight), if they have, say, 4-5 as hole cards and Q-6-7-8 on the poker table, then four 3s and four 9s become the total 8 possible outs available for the river round to complete the Straight.

Hence, the number of ways to improve hand strength here is 8. Now, let’s count the number of ways hand strength will not improve. Out of 52 cards, 2 are hole cards, 4 are on the table, and 8 are the outs. Hence, the remaining 38 cards will not improve the odds in the river.

• Hence, the ratio of odds not improving to the ratio of odds improving here will be 38:8, 4.75:1. This ratio indicates that for every 1 time we improve, every 4.75 times we don’t. So, during the game, if we get such a situation and we have to call, in such case, we need to have pot odds greater than 4.75 to 1

Calculating Pot Odds

Pot Odds = existing ₹ in the main pot+₹ added in the pot during the current betting round/₹ to pay during a call.

Consider this scenario: The initial pot holds ₹100, and another ₹100 is thrown into the mix. It brings the total pot value to ₹200, and you are faced with the decision to call a bet of ₹100.

In this situation, the pot odds are effectively ₹200 to ₹100, translating to odds of 2 to 1.

For another example, let's assume there is ₹ 200 in the main pot. In the betting round, the first player bets ₹ 100, the second player calls ₹ 100, and next is your turn to act.

What are your pot odds?

Using the formula above, we arrive at the following equation: ₹400: ₹100 or 4 to 1.

(₹200 in the main pot) + (₹200 from this betting round)/ (₹100 that you have to pay (call))

So, to make a profitable call, we require pot odds greater than 4.75 to 1 (i.e. ₹500 already in the pot against a bet of 100).

Basic Probability Rules Poker

It is practically impossible to keep referring to the poker probability chart while calculating the possibility of a hand forming. So there are some basic quick calculation rules which players can use while being in a game:

• Once the flop is open, you can find the possibility of a hand to be formed by turn or river by multiplying the number of outs by 4. For example, if your hole cards are 2-3 unsuited and the flop opens to be 4-5-8, then the possibility of a straight getting formed is calculated by multiplying 4 by the number of outs. Here, the outs will be the cards that will complete the straight hand if opened in either the turn or the river round. In this case, there are 8 outs, which are A-A-A-A/6-6-6-6. Hence, the possibility of a straight forming by the turn or river round is 32%.

• To calculate the possibility of a hand to be formed by the next step, multiply the number of outs for that hand by the number of 2.

• To get the best value of the hand, players have to adjust the number of outs at times if their opponents might be holding some of those outs. For example, if a player holds two diamonds, the flop opens up with 2 cards of diamonds and one heart. Let’s say the flop has a pair among the three cards. Now, the player is expecting a flush draw, which makes their outs 9, whereas, with a pair on the board, chances are that some other player might be holding a full house. So, the player with the flush draw should relook at their strategy and calculate the odds by reducing the number of outs, let’s say by reducing it from 9 to 8, thereby addressing caution.

What If I Wanted To Practice Poker Probability Myself?

Practicing poker probability calculation is a useful skill in poker. Let’s understand this with an example:

Flush Draw Example:

Suppose you have two spade cards as hole cards, and let’s say the flop also contains two spades. So, there is a situation where there is a flush draw.

1. Now, the possibility of completing a flush by the turn round will be calculated as the ratio of the number of outs possible divided by the number of unknown cards. So, it is 9/47 = 19.15%.
2. Similarly, the odds of completing a flush in the river round will be calculated as (39/47)*(9/46) = 16.23%.
3. Therefore, the odds of getting flush completed in either the turn or the river round = 19.15% + 16.23% = 35.38%.

As we saw earlier, this result is similar to the shortcut of multiplying the number of outs by 4 if we are trying to calculate the odds of completing a hand by the river round.

Frequently Asked Questions

How does probability work in poker?

Probability helps you understand the odds of drawing a winning hand or navigating your betting decisions based on your opponent's cards.

How do you calculate probability in poker?

To calculate your poker probability, you must consider your cards, the community cards, and the number of opponents you are playing against. You can then use mathematical formulas to determine your probability of winning the hand.

Is poker based on probability?

Yes, poker is based on probability. Players use the odds of getting certain hands to make strategic decisions. Understanding probability involves a significant amount of skill.

What is the formula for probability in gambling?

The formula for probability is: probability (p) equals the total number of favourable outcomes (f) divided by the total number of possibilities (t) or p=f/t.

Conclusion

Understanding the poker probabilities is important for making informed decisions in the game. It helps players understand the odds of getting specific hands and how likely they are to win against their opponents. Each poker variant has its own set of probabilities, but in general, the stronger your hand and the fewer opponents you have, the better your chances of winning. Players can enhance their strategies by learning to calculate these probabilities and boost their success at the poker table.