Poker Probability Calculator

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However, you have a shortcut to apply correct poker math on the go and quickly calculate the probability of hitting your hand while playing. Therefore, I will explain a simple rule, which will help you a lot with Texas Holdem odds. There is quite an easy way to get the answer very fast and we can call it “the rule of 2 and 4”. The Which Poker Hand Wins Calculator. Try our winning hand generator below and you'll soon master the mystery of 'Which hand wins?' If you need some help figuring out just how to use the poker hand calculator, there's a simple guide and frequently asked questions below. See More Online Poker Sites.

Online poker odds calculators are poker software tools that automatically calculate the poker odds for a hand, using the information that is available. These poker statistics that the poker odds calculators generate are especially useful as a tool for beginner poker players who are not completely familiar with the odds of making different poker hands. Advanced poker players will be more interested in products such as Holdem Ranger that allowed for detailed calculations of equity using hand ranges. If you are interested in extremely detailed mathematical projections of poker situations and aren't afraid to get elbow deep in formulas then check out Pokerazor. Simple Free Poker Calculators can be found on sites like PocketFives.com and CardPlayer.com.
Most poker calculators will display your odds of making a variety of hands such as straights, flushes, full houses, etc. as well as your opponents' odds of having those hands. In most poker odds calculators, all of these statistics are combined to give you an estimated win percentage as well as a recommended action. Some poker odds calculators run in real time and read the cards that are on the table to gather the data. Others, such as PokerStove, are used on their own and require the user to enter all of the hand information.
When deciding what kind of Hold'em calculator you want to use, take into consideration what features you would like to have. If you are interested in getting probabilities and only play a single table, check out the Poker Wingman poker odds calculator. It attached to a single table and offers odds and advice based on customizable playing style settings. If you want to go through and analyze hands after they have been played to find out if you put your money in correctly vs. your opponent's range of hands, check out PokerStove or Hold'em Ranger. For a solid free poker odds calculator, check out Magic Holdem.
Poker odds calculators are extremely useful tools when used correctly. Do not make the mistake of choosing one that gives poor advice or incorrect statistics. Some calculators are banned at certain poker sites. This is usually because you are not allowed to use programs that give advice in real time. Another reason that online poker calculators such as Poker-edge are banned is because they used a central database of hand histories to calculate opponent's playing styles and makes recommendations based on that.

An excellent odds calculator that attaches itself to your table automatically and gives you helpful statistics about the current hand as well as past hands against your opponents.

5.00 out of 5

A great Omaha calculator and tracking program from the folks who brought us Tournament Indicator.

5.00 out of 5

A downloadable odds calculator that allows quick computation and analysis.

5.00 out of 5

Excellent odds calculator that requires no setup time

3.83 out of 5

Review of poker software applications for the Apple iPhone

3.54 out of 5

No Video Available

Master flop textures and hand ranges with Flopzilla. The program will allow you to sharpen your range-assigning skills so that they become second nature.

Poker Probability Calculations

3.30 out of 5

Online poker odds calculator and HUD for the Mac

3.00 out of 5

Review of Stud Indicator, which keeps tabs on you and your opponents at Stud tables.

Not Rated Yet

Review of Ace Poker Drills, which consists of an Equity Trainer, Pre-Flop Trainer, and an Odd/Outs and Equity Calculator.

Not Rated Yet

Covers all of your needs for basic and advanced ICM calculations

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A mobile poker odds calculator

Not Rated Yet

No Video Available

Poker odds calculator available in-game or during study sessions

Not Rated Yet

Poker calculator for Hold'em, Omaha, Razz, Stud, and Stud Eight or Better

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One of the first online Omaha poker odds calculators dedicated to Mac poker players.

Not Rated Yet

Review of Flop Inspector and Flop Inspector Range Visualizers

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Review of Poker Odds – Apple Watch Edition, which provides odds and equity on your Apple Watch

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Introduction

Derivations for Five Card Stud

I have been asked so many times how I derived the probabilities of drawing each poker hand that I have created this section to explain the calculation. This assumes some level mathematical proficiency; anyone comfortable with high school math should be able to work through this explanation. The skills used here can be applied to a wide range of probability problems.

The Factorial Function

If you already know about the factorial function you can skip ahead. If you think 5! means to yell the number five then keep reading.

The instructions for your living room couch will probably recommend that you rearrange the cushions on a regular basis. Let's assume your couch has four cushions. How many combinations can you arrange them in? The answer is 4!, or 24. There are obviously 4 positions to put the first cushion, then there will be 3 positions left to put the second, 2 positions for the third, and only 1 for the last one, or 4*3*2*1 = 24. If you had n cushions there would be n*(n-1)*(n-2)* ... * 1 = n! ways to arrange them. Any scientific calculator should have a factorial button, usually denoted as x!, and the fact(x) function in Excel will give the factorial of x. The total number of ways to arrange 52 cards would be 52! = 8.065818 * 10⁶⁷.

The Combinatorial Function

Assume you want to form a committee of 4 people out of a pool of 10 people in your office. How many different combinations of people are there to choose from? The answer is 10!/(4!*(10-4)!) = 210. The general case is if you have to form a committee of y people out of a pool of x then there are x!/(y!*(x-y)!) combinations to choose from. Why? For the example given there would be 10! = 3,628,800 ways to put the 10 people in your office in order. You could consider the first four as the committee and the other six as the lucky ones. However you don't have to establish an order of the people in the committee or those who aren't in the committee. There are 4! = 24 ways to arrange the people in the committee and 6! = 720 ways to arrange the others. By dividing 10! by the product of 4! and 6! you will divide out the order of people in an out of the committee and be left with only the number of combinations, specifically (1*2*3*4*5*6*7*8*9*10)/((1*2*3*4)*(1*2*3*4*5*6)) = 210. The combin(x,y) function in Excel will tell you the number of ways you can arrange a group of y out of x.

Now we can determine the number of possible five card hands out of a 52 card deck. The answer is combin(52,5), or 52!/(5!*47!) = 2,598,960. If you're doing this by hand because your calculator doesn't have a factorial button and you don't have a copy of Excel, then realize that all the factors of 47! cancel out those in 52! leaving (52*51*50*49*48)/(1*2*3*4*5). The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability.

Poker Math

The next section shows how to derive the number of combinations of each poker hand in five card stud.

Royal Flush

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There are four different ways to draw a royal flush (one for each suit).

Straight Flush

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight flushes.

Four of a Kind

There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 48. Thus there are 13 * 48 = 624 different four of a kinds.

Full House

There are 13 different possible ranks for the three of a kind, and 12 left for the two of a kind. There are 4 ways to arrange three cards of one rank (4 different cards to leave out), and combin(4,2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a full house.

Flush

There are 4 suits to choose from and combin(13,5) = 1,287 ways to arrange five cards in the same suit. From 1,287 subtract 10 for the ten high cards that can lead a straight, resulting in a straight flush, leaving 1,277. Then multiply for 4 for the four suits, resulting in 5,108 ways to form a flush.

Straight

The highest card in a straight can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible high cards. Each card may be of four different suits. The number of ways to arrange five cards of four different suits is 4⁵ = 1024. Next subtract 4 from 1024 for the four ways to form a flush, resulting in a straight flush, leaving 1020. The total number of ways to form a straight is 10*1020=10,200.

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Three of a Kind

There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are combin(12,2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus the number of ways to arrange a three of a kind is 13 * 4 * 66 * 4² = 54,912.

Two Pair

There are (13:2) = 78 ways to arrange the two ranks represented. In both ranks there are (4:2) = 6 ways to arrange two cards. There are 44 cards left for the fifth card. Thus there are 78 * 6² * 44 = 123,552 ways to arrange a two pair.

One Pair

There are 13 ranks to choose from for the pair and combin(4,2) = 6 ways to arrange the two cards in the pair. There are combin(12,3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 4³ = 1,098,240 ways to arrange a pair.

Nothing

First find the number of ways to choose five different ranks out of 13, which is combin(13,5) = 1287. Then subtract 10 for the 10 different high cards that can lead a straight, leaving you with 1277. Each card can be of 1 of 4 suits so there are 4⁵=1024 different ways to arrange the suits in each of the 1277 combinations. However we must subtract 4 from the 1024 for the four ways to form a flush, leaving 1020. So the final number of ways to arrange a high card hand is 1277*1020=1,302,540.

Specific High Card

For example, let's find the probability of drawing a jack-high. There must be four different cards in the hand all less than a jack, of which there are 9 to choose from. The number of ways to arrange 4 ranks out of 9 is combin(9,4) = 126. We must then subtract 1 for the 10-9-8-7 combination which would form a straight, leaving 125. From above we know there are 1020 ways to arrange the suits. Multiplying 125 by 1020 yields 127,500 which the number of ways to form a jack-high hand. For ace-high remember to subtract 2 rather than 1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and 5-4-3-2-A are both valid straights. Here is a good site that also explains how to calculate poker probabilities.

Five Card Draw — High Card Hands

Hand	Combinations	Probability
Ace high	502,860	0.19341583
King high	335,580	0.12912088
Queen high	213,180	0.08202512
Jack high	127,500	0.04905808
10 high	70,380	0.02708006
9 high	34,680	0.01334380
8 high	14,280	0.00549451
7 high	4,080	0.00156986
Total	1,302,540	0.501177394

Ace/King High

For the benefit of those interested in Caribbean Stud Poker I will calculate the probability of drawing ace high with a second highest card of a king. The other three cards must all be different and range in rank from queen to two. The number of ways to arrange 3 out of 11 ranks is (11:3) = 165. Subtract one for Q-J-10, which would form a straight, and you are left with 164 combinations. As above there 1020 ways to arrange the suits and avoid a flush. The final number of ways to arrange ace/king is 164*1020=167,280.

Internal Links

Poker Probabilities

For lots of other probabilities in poker, please see my section on Probabilities in Poker.

6 Card Poker Probability Calculator

Written by:Michael Shackleford

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