1. Poker Hand Combinations Math Worksheets
2. Poker Hand Combinations Math Manipulatives
3. Poker Hand Combinations Math Solver
4. Poker Hand Combinations Math Games

This way you will have all possibilities, with no duplicates in the hands or within hands of players. (player one having the hand of player two in a different deal and vice versa) I hope this helped you, or someone else. I'm not giving code examples, because this question is more like a how to solve this problem, and the others gave you coding. Solution for Poker Hands Using combinations, calculate the number of each poker hand in a deck of cards. (A poker hand consists of 5 cards dealt in any order.).

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?

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

Poker Hand Combinations Math Worksheets

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.

One plays poker with a deck of 52 cards, which come in 4 suits (hearts, clubs, spades, diamonds) with 13 values per suit (A, 2, 3, …, 10, J, Q, K).

In poker one is dealt five cards and certain combinations of cards are deemed valuable. For example, a “four of a kind” consists of four cards of the same value and a fifth card of arbitrary value. A “full house” is a set of three cards of one value and two cards of a second value. A “flush” is a set of five cards of the same suit. The order in which one holds the cards in ones hand is immaterial.

Answer: Again this is a multi-stage problem with each stage being its own separate labeling problem. One way to help tease apart stages is to image that you’ve been given the task of writing a computer program to create poker hands. How will you instruct the computer to create a flush?

First of all, there are four suits – hearts, spades, clubs and diamonds – and we need to choose one to use for our flush. That is, we need to label one suit as “used” and three suits as “not used.” There are (dfrac{4!}{1!3!} = 4) ways to do this.

Second stage: Now that we have a suit, we need to choose five cards from the 13 cards of that suit to use for our hand. Again, this is a labeling problem – label five cards as “used” and eight cards as “not used.” There are (dfrac{13!}{5!8!} = 1287) ways to do this.

By the multiplication principle there are (4 times 1287 = 5148) ways to compete both stages. That is, there are (5148) possible flushes.

COMMENT: There are (dfrac{52!}{5!47!} = 2598960) five-card hands in total in poker. (Why?) The chances of being dealt a flush are thus: (dfrac{5148}{2598960} approx 0.20%).

Answer: This problem is really a three-stage labeling issue.

First we must select which of the thirteen card values – A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K – is going to be used for the triple, which will be used for the double, and which 11 values are going to be ignored. There are (dfrac{13!}{1!1!11!} = 13 times 12 = 156) ways to accomplish this task.

Among the four cards of the value selected for the triple, three will be used for the triple and one will be ignored. There are (dfrac{4!}{3!1!} = 4) ways to accomplish this task. Among the four cards of the value selected for the double, two will be used and two will be ignored. There are (dfrac{4!}{2!2!} = 6) ways to accomplish this.

By the multiplication principle, there are (156 times 4 times 6 = 3744) possible full houses.

COMMENT: High-school teacher Sam Miskin recently used this labeling method to count poker hands with his high-school students. To count how many “one pair hands” (that is, hands with one pair of cards the same numerical value and three remaining cards each of different value) he found it instructive bring 13 students to the front of the room and hand each student four cards the same value from a single deck of cards.

He then asked the remaining students to select which of the thirteen students should be the “pair” and which three should be the “singles.” He had the remaining nine students return to their seats.

He then asked the “pair” student to raise his four cards in the air and asked the seated students to select which two of the four should be used for the pair. He then asked each of the three “single” students in turn to hold up their cards while the seated students selected on one the four cards to make a singleton.

This process made the multi-stage procedure clear to all and the count of possible one pair hands, namely,

(dfrac{13!}{1!3!9!} times dfrac{4!}{2!2!} times 4 times 4 times 4)

readily apparent.

Poker Hand Combinations Math Manipulatives

Answer: A straight can begin with A, 2, 3, 4, 5, 6, 7, 8, 9 or 10. We must first select which of these values is to be the start of our straight. There are 10 choices.

For the starting value we must select which of the four suits it will be. There are 4 choices.

There are also 4 choices for the suit of the second card in the straight, 4 for the third, 4 for the fourth, and 4 for the fifth.

By the multiplication principle, the total number of straights is:

Poker Hand Combinations Math Solver

Poker Hand Combinations Math Games

(10 times 4 times 4 times 4 times 4 times 4 = 10240).

The chances of being dealt a straight are about 0.39%.