Poker Starting Hands Probability

The main underpinning of poker is math – it is essential. For every decision you make, while factors such as psychology have a part to play, math is the key element.

In this lesson we’re going to give an overview of probability and how it relates to poker. This will include the probability of being dealt certain hands and how often they’re likely to win. We’ll also cover how to calculating your odds and outs, in addition to introducing you to the concept of pot odds. And finally we’ll take a look at how an understanding of the math will help you to remain emotional stable at the poker table and why you should focus on decisions, not results.

What is Probability?

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Probability is the branch of mathematics that deals with the likelihood that one outcome or another will occur. For instance, a coin flip has two possible outcomes: heads or tails. The probability that a flipped coin will land heads is 50% (one outcome out of the two); the same goes for tails.

Probability and Cards

When dealing with a deck of cards the number of possible outcomes is clearly much greater than the coin example. Each poker deck has fifty-two cards, each designated by one of four suits (clubs, diamonds, hearts and spades) and one of thirteen ranks (the numbers two through ten, Jack, Queen, King, and Ace). Therefore, the odds of getting any Ace as your first card are 1 in 13 (7.7%), while the odds of getting any spade as your first card are 1 in 4 (25%).

Unlike coins, cards are said to have “memory”: every card dealt changes the makeup of the deck. For example, if you receive an Ace as your first card, only three other Aces are left among the remaining fifty-one cards. Therefore, the odds of receiving another Ace are 3 in 51 (5.9%), much less than the odds were before you received the first Ace.

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Pre-flop Probabilities: Pocket Pairs

In order to find the odds of getting dealt a pair of Aces, we multiply the probabilities of receiving each card:

(4/52) x (3/51) = (12/2652) = (1/221) ≈ 0.45%.

To put this in perspective, if you’re playing poker at your local casino and are dealt 30 hands per hour, you can expect to receive pocket Aces an average of once every 7.5 hours.

The odds of receiving any of the thirteen possible pocket pairs (twos up to Aces) is:

(13/221) = (1/17) ≈ 5.9%.

In contrast, you can expect to receive any pocket pair once every 35 minutes on average.

Pre-Flop Probabilities: Hand vs. Hand

Players don’t play poker in a vacuum; each player’s hand must measure up against his opponent’s, especially if a player goes all-in before the flop.

Here are some sample probabilities for most pre-flop situations:

Post-Flop Probabilities: Improving Your Hand

Now let’s look at the chances of certain events occurring when playing certain starting hands. The following table lists some interesting and valuable hold’em math:

Many beginners to poker overvalue certain starting hands, such as suited cards. As you can see, suited cards don’t make flushes very often. Likewise, pairs only make a set on the flop 12% of the time, which is why small pairs are not always profitable.

PDF Chart

We have created a poker math and probability PDF chart (link opens in a new window) which lists a variety of probabilities and odds for many of the common events in Texas hold ‘em. This chart includes the two tables above in addition to various starting hand probabilities and common pre-flop match-ups. You’ll need to have Adobe Acrobat installed to be able to view the chart, but this is freely installed on most computers by default. We recommend you print the chart and use it as a source of reference.

Odds and Outs

If you do see a flop, you will also need to know what the odds are of either you or your opponent improving a hand. In poker terminology, an “out” is any card that will improve a player’s hand after the flop.

One common occurrence is when a player holds two suited cards and two cards of the same suit appear on the flop. The player has four cards to a flush and needs one of the remaining nine cards of that suit to complete the hand. In the case of a “four-flush”, the player has nine “outs” to make his flush.

A useful shortcut to calculating the odds of completing a hand from a number of outs is the “rule of four and two”. The player counts the number of cards that will improve his hand, and then multiplies that number by four to calculate his probability of catching that card on either the turn or the river. If the player misses his draw on the turn, he multiplies his outs by two to find his probability of filling his hand on the river.

In the example of the four-flush, the player’s probability of filling the flush is approximately 36% after the flop (9 outs x 4) and 18% after the turn (9 outs x 2).

Pot Odds

Another important concept in calculating odds and probabilities is pot odds. Pot odds are the proportion of the next bet in relation to the size of the pot.

For instance, if the pot is $90 and the player must call a $10 bet to continue playing the hand, he is getting 9 to 1 (90 to 10) pot odds. If he calls, the new pot is now $100 and his $10 call makes up 10% of the new pot.

Experienced players compare the pot odds to the odds of improving their hand. If the pot odds are higher than the odds of improving the hand, the expert player will call the bet; if not, the player will fold. This calculation ties into the concept of expected value, which we will explore in a later lesson.

Bad Beats

A “bad beat” happens when a player completes a hand that started out with a very low probability of success. Experts in probability understand the idea that, just because an event is highly unlikely, the low likelihood does not make it completely impossible.

A measure of a player’s experience and maturity is how he handles bad beats. In fact, many experienced poker players subscribe to the idea that bad beats are the reason that many inferior players stay in the game. Bad poker players often mistake their good fortune for skill and continue to make the same mistakes, which the more capable players use against them.

Decisions, Not Results

One of the most important reasons that novice players should understand how probability functions at the poker table is so that they can make the best decisions during a hand. While fluctuations in probability (luck) will happen from hand to hand, the best poker players understand that skill, discipline and patience are the keys to success at the tables.

A big part of strong decision making is understanding how often you should be betting, raising, and applying pressure.
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Conclusion

A strong knowledge of poker math and probabilities will help you adjust your strategies and tactics during the game, as well as giving you reasonable expectations of potential outcomes and the emotional stability to keep playing intelligent, aggressive poker.

Remember that the foundation upon which to build an imposing knowledge of hold’em starts and ends with the math. I’ll end this lesson by simply saying…. the math is essential.

By Gerald Hanks

Gerald Hanks is from Houston Texas, and has been playing poker since 2002. He has played cash games and no-limit hold’em tournaments at live venues all over the United States.

Poker is a very technical casino game and you can easily be overwhelmed if you are just starting out. There are different variations of online poker games each with slightly different rules.

Poker Starting Hands Odds Of Winning

However, the ten different types of hands you can make are constant.

Finding an easy way for a beginner to learn these hands will help you get started quickly. With this poker hand cheat sheet, you’ll get an overview of all the poker hands you can make.

While there will be slight variations as you play different poker games, this should not deter you from getting started with the basics.

Highest value hands in Poker – the Flushes

Each hand in poker is made up of five cards. Depending on the game variant, players can make that hand from different numbers of their own and community cards.

Poker hands are ranked in value and the rules that govern the content of each hand are the same:

• The highest value hand is the Royal Flush – a straight run of five cards with values from 10 to Ace all in the same suit.
• Straight Flush is the next highest – all cards are from the same suit and run sequentially, e.g. 6 to 10 of spades or 9 to K of diamonds.

Middle value hands in poker

After the Flushes, there are several other hands that you can form to hit a win in poker. In order of value, they are:

• Four of a Kind – which involves four cards of the same rank but different suits.
• A Full House – which is a combination of three of a kind and a pair.
• A Flush – five cards from the same suit, not in any sequence.
• A Straight – five cards in a sequence, but not from the same suit.
• Three of a Kind – three cards with the same rank, but from different suits.

Lowest value hands in poker

The last three poker hands are the lowest-ranked and are:

• Two pair – Two different pairs with all cards being from any suit.
• Pair – two cards of the same rank but different suits.
• High card – This is only used when you have not made any of the hands previously described.

In this case, the highest card in your hand is played. One other point to remember is that if you play the same hand as the dealer, the value of the cards within that hand determines the winner.

For example, if both have four-of-a-kind, then four kings would win over four sixes.

Poker hand odds

To successfully apply what you have learnt, you need to understand the odds of different hands in various poker games.

Generally, poker hands are played with 5 cards, however, there are some variants that use 7 cards. These games have higher odds for making specific hands compared to 5-card poker, although not in all cases.

For a Royal flush, a 5-card poker game has odds of this occurring of 0.00015% compared to 0.0032% in a 7-card poker game.

However, the odds of getting a high card are 50.1% in a 5-card poker game compared to 17.4% in 7-card poker.

Using Poker tools

There are different forms of poker tools you can use to improve your skill. The most popular type is the poker simulator.

This tool mimics a real poker round, however, you will be playing against a computerized dealer.

The good thing is that it is free. Another poker tool is the Propoker tool. This is an equity calculator for various versions of poker games.

If you want a tool made for beginners, you can check out the Hold’em Poker trainer. This simple tool provides you with free online exercises to practice ranking hands, making the best hand, and more.

Probability In Poker

Poker Hand Cheat Sheet Conclusion

Calculate Poker Hands Probability

This poker hand cheat sheet is going to be helpful when playing in the real world of poker but there is one other important thing to bear in mind. That is positional play, which relates to your position at the table in relation to the dealer.

The three positions are termed early, middle, and late.

The early position is usually considered to be the first three seats to the left of the dealer and is considered weak. The middle position is considered better since it offers the opportunity to play based on what the early position players have. The late position is the strongest as you’ll have seen all other players’ actions.

However, when playing poker online, you play against the dealer only, regardless of whether you are playing virtual or live poker games.

Poker Starting Hands Probability Calculator

For beginners, having a good grasp of the different poker hands and their respective rankings will stand you in good stead. This is where our poker hand cheat sheet will come in handy and help you get ahead.

Of course, the more you play the more familiar you’ll become with the different hands. Then you can test out your skill for free using some of the online poker tools available, before playing for real money at your favorite casino site.

FAQs

The top 10 hands:
1. Royal flush – A, K, Q, J, 10, all the same suit.
2. Straight flush – Five sequenced cards in the same suit.
3. Four of a kind – Four cards with the same rank.
4. Full house – Any three of a kind with a pair.
5. Flush – Five cards with the same suit that’s not in a sequence.
6. Straight – Five sequenced cards that’s not of the same suit.
7. Three of a kind – Any three cards with the same rank.
8. Two pair – Two pairs of different ranks and suits.
9. Pair – Two cards with the same rank.
10. High Card – None of the above but you have the highest card.

Poker Starting Hands Probability

It’s an Ace-High Royal Flush or Ace, King, Queen, Jack, Ten with the same suit.

Poker Starting Hands Probability Problems

Any big pocket pair, high cards like Ace, King, Queen, and Jack. Always play big-suited connectors and big-connectors off suit.