Rate Poker Hands
2021年12月9日Register here: http://gg.gg/x6sj7
*What Is The Best Poker Hands
*Rate Poker Hands
*How To Rate Poker Hands
*Rate Poker Hands
Until you find Rate Poker Handsyour favorite game.Demo mode is when you Rate Poker Handscan try the Rate Poker Handsgame for free with play money, you Rate Poker Handscan play Rate Poker Handscasino games as long as you want, all free. By using the casino games in demo mode, you have the.
Knowing your hands per hour is important for planning your sessions and figuring out your hourly rate. I dove into the statistics and did a bit of research and this is what I found out.
How many hands per hour in poker? Online you will play around 75 hands per hour on a 6-max table and about 62 hands per hour on a 9-max table. In online fast fold games like Zoom, Zone, or Snap you will play 200 to 250 hands per table hour. Live poker players will typically average 25 hands per hour.
Nfl vs xfl. Of course, these numbers can vary based on where you are playing. Let’s take a look at the factors that can effect your hands per hour experience.There Are Fewer Hands per Hour in Soft Games
As the VPIP goes up the hands per hour tends to go down. For comparison sake, let’s take a look at Bovada/Ignition versus the Winning Poker Network (WPN). I have played hundreds of thousands of hands on both networks and checked my database to compare the numbers.
*The poker odds calculators on CardPlayer.com let you run any scenario that you see at the poker table, see your odds and outs, and cover the math of winning and losing poker hands. Texas Hold’em Omaha.
*Poker Starting Hands - Comprehensive guide to which poker hands you should play, including a 2021 Texas Hold’em poker starting hands chart.
*The current universal standard used to measure a poker player’s win rate is how many big blinds they win per 100 hands played. The shorthand version used to express the win rate is to state the number of big blinds won followed by bb/100. For example, a player might say, “my win rate over my past 100,000 hands at 50NL has been 4bb/100.”.
For 6-max games on Bovada and Ignition I average 68 hands per hour. On WPN I average 77 hands per hour. In 9-max games the numbers are 55 and 62 respectively.
Why the discrepancy?
It’s because the VPIP is quite a bit higher on Bovada and Ignition. I mean it makes sense. If more people are limping and seeing flops then, of course, the games will play slower. I am not a live player but I can imagine 5/10 players enjoy a higher hands per hour than 1/2 players.What Are Your Exact Hands per Hour on Each Site?
Hold’em Manager has a handy multi-table report that shows exactly how many hands per hour are played based on how many tables are played at once. Here are my numbers:6-MaxPoker Network6-Max Hands per HourIgnition/Bovada68.47Winning Poker Network76.53
9-MaxPoker Network9-Max Hands per HourIgnition/Bovada55.36Winning Poker Network61.89
Surprisingly, the more tables I played the higher my hands per hour is. I guess this makes perfect sense since game selection pretty much goes out the window when you are mass multi-tabling.Hands per Hour Dealt Is Different Than Hands per Hour Played
If you look around the web you will see much higher numbers than I am writing about here. That’s because they are talking about hands per hour when you are actually dealt into the hand.
Raw hands per hour when actually being dealt in is like 90 per hour for 6-max and 75 per hour for 9-max. However, unless you are staying on the same table and playing for hours, lower numbers are more realistic.
What they fail to take into account that some of the time that you are sitting at a table you are not being dealt in. Hands dealt per hour does not include time players spend getting on and off the tables and sitting and waiting on big blinds. Therefore, actual hands played per hour is what you should use when trying to figure out your hourly win-rate.What About Hands Per Hour in Live Games?
Live dealt in numbers tend to be much closer to reality since there is no multi-tabling and a player will typically stay at the same table for hours. Even so, the number of hands played per hour is still much lower than online.
Hands per hour in live games can range anywhere from 20 to 30. Besides almost exclusively being 9 or 10 handed, it takes time for the dealer to shuffle and deal. That leaves little time to actually playing poker. I have a ton of respect for the patience and mental toughness of live grinders. Aggressive Table Selection Affects Hands per Hour
The longer you stay on a table the more hands per hour you are dealt. This is because you have to pull up a new table, sit down, and then wait for the big blinds. This takes time. Therefore, switching games often lowers your hands per hour.
Now I am not saying you should stop game selecting. In today’s tough poker environment, robust table selection is how money is made. One way to remedy this issue and still maintain your desired number of hands per hour is to just add one more table. If you normally two table, try three.Mass Multi-Tablers Slow down the GamesWhat Is The Best Poker Hands
I have noticed a direct correlation between the maximum number of tables allowed on a site to how many hands per hour is played. This is due to mass multi-tablers, who tend to time bank often and hold up the games.
For example, Bovada and Ignition also only allow 4 cash game tables to be played at once. WPN still allows people to play 24 tables at the same time. Get two or three guys playing 24 tables on one table and it becomes time bank city. Watch your hourly rate shrink due to atrophied hands per hour.Why Does Full Ring Play Slower Than 6-Max?
Full ring has a lower hands per hour because there are more players on the table that have to act during each hand. Also, 9-max games tend to play more passively than 6-max. Therefore, 6-max games tend to play about 20% faster than 9-max games.What If I Mix in Full Ring Games and Want to Have the Same Hands per Hour?
If you want to play the same amount of hands per hour at 9-max that you do at 6-max you will simply just need to add more tables.Rate Poker Hands
Let’s say you currently play 8 tables of 6-max and average 600 hands per hour. Since the average hands per hour at 9-max is 62, you simply divide 62 into 600 and that gives you your optimal number of full ring tables.
In this case, it is 9.68, or 10 tables.
*600/62= 9.68Fast-Fold (Zoom/Zone) Poker Hands Per Hour
In the last few years, the popularity of fast-fold poker has increased substantially due to the increase in hands per hour attainable. No longer do you have to sit back and watch a hand play out after you have folded. With the click of a button you are moved on to another hand.
This allows people to play between 200 and 300 hands per hour on each table.
What this means is that a player can use less real estate on his or her desktop and just focus on one or two tables at the time, while still achieving a substantial number of hands per hour.Drawbacks of Fast-Fold Poker
Fast-fold poker is not for the feint of heart. In fact, it takes a completely different strategy than you would play on a regular table. Maximizing hands per hour can actually be a big factor in determining how to play.
Personally, I dislike fast-fold poker not because of the strategic differences or the speed of play. Rather, I believe that the game is not really poker at all.
Observing opponents and then adjusting your strategy based on what they do is at the heart of poker. Fast-fold poker takes this away since every hand will find you in a different mix of opponents. So much for using gameflow considerations to your advantage.Final Thoughts
If you are reading this article so that you can start mass multi-tabling because you think it will increase your hourly rate substantially, I have to warn you. As hands per hour increases, ability to focus on what is going on and game select correctly tends to go out the window.
So unless you are at least a 6bb/100 winner playing one or two tables, I would not even consider trying to maximize your hands per hour yet. Your growth as a poker player is much more important than trying to increase your hourly rate.
However, once you are ready you now know how many hands per hour you can expect to play at the tables. I hope this information allows you to figure out your optimal number of tables to play to achieve your hourly rate goals. My optimal number of tables is about 6, what’s yours?Related Questions
How do I calculate hourly rate in poker?Tocalculate your hourly rate you divide your total profit by the number of hours you have played.
What is a good poker hourly rate? What can be considered a good hourly rate in poker will vary from player to player based on their goals. It will also depend on the stake being played. It is much better to focus on win-rate instead of hourly rate. A good online win-rate will range from 2-4 big blinds per 100 hands for multi-tablers and from 4 big blinds to 10 big blinds per 100 hands for single tablers. However, due to the extreme softness of some live games, live players will often enjoy a win-rate much higher.
How many hands per hour in tournament poker? Hands per hour in tournaments will be higher than cash games since there is no table selection involved. Online you can expect to play 90 hands per hour for 6-max and 75 hands per hour for 9-max. As the blinds increase, hands also tend to play faster since the game evolves into push-fold. Live tournament players play around 30 hands per hour.
This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities
Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.
___________________________________________________________________________Preliminary Calculation
Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.These are the same hand. Order is not important.
The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.
The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.
Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is
This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.
The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.
If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.
Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.
Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:
One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.
___________________________________________________________________________How To Rate Poker HandsThe Poker Hands
Here’s a ranking chart of the Poker hands.
The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.Rate Poker Hands
Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.
The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.
Definitions of Poker HandsPoker HandDefinition1Royal FlushA, K, Q, J, 10, all in the same suit2Straight FlushFive consecutive cards,all in the same suit3Four of a KindFour cards of the same rank,one card of another rank4Full HouseThree of a kind with a pair5FlushFive cards of the same suit,not in consecutive order6StraightFive consecutive cards,not of the same suit7Three of a KindThree cards of the same rank,2 cards of two other ranks8Two PairTwo cards of the same rank,two cards of another rank,one card of a third rank9One PairThree cards of the same rank,3 cards of three other ranks10High CardIf no one has any of the above hands,the player with the highest card wins
___________________________________________________________________________Counting Poker Hands
Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.
Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.
Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is
Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?
Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.
Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.
Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.
Two Pair and One Pair
These two are left as exercises.
High Card
The count is the complement that makes up 2,598,960.
The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.
Probabilities of Poker HandsPoker HandCountProbability2Straight Flush400.00001543Four of a Kind6240.00024014Full House3,7440.00144065Flush5,1080.00196546Straight10,2000.00392467Three of a Kind54,9120.02112858Two Pair123,5520.04753909One Pair1,098,2400.422569010High Card1,302,5400.5011774Total2,598,9601.0000000
___________________________________________________________________________
2017 – Dan Ma
Register here: http://gg.gg/x6sj7
https://diarynote.indered.space
*What Is The Best Poker Hands
*Rate Poker Hands
*How To Rate Poker Hands
*Rate Poker Hands
Until you find Rate Poker Handsyour favorite game.Demo mode is when you Rate Poker Handscan try the Rate Poker Handsgame for free with play money, you Rate Poker Handscan play Rate Poker Handscasino games as long as you want, all free. By using the casino games in demo mode, you have the.
Knowing your hands per hour is important for planning your sessions and figuring out your hourly rate. I dove into the statistics and did a bit of research and this is what I found out.
How many hands per hour in poker? Online you will play around 75 hands per hour on a 6-max table and about 62 hands per hour on a 9-max table. In online fast fold games like Zoom, Zone, or Snap you will play 200 to 250 hands per table hour. Live poker players will typically average 25 hands per hour.
Nfl vs xfl. Of course, these numbers can vary based on where you are playing. Let’s take a look at the factors that can effect your hands per hour experience.There Are Fewer Hands per Hour in Soft Games
As the VPIP goes up the hands per hour tends to go down. For comparison sake, let’s take a look at Bovada/Ignition versus the Winning Poker Network (WPN). I have played hundreds of thousands of hands on both networks and checked my database to compare the numbers.
*The poker odds calculators on CardPlayer.com let you run any scenario that you see at the poker table, see your odds and outs, and cover the math of winning and losing poker hands. Texas Hold’em Omaha.
*Poker Starting Hands - Comprehensive guide to which poker hands you should play, including a 2021 Texas Hold’em poker starting hands chart.
*The current universal standard used to measure a poker player’s win rate is how many big blinds they win per 100 hands played. The shorthand version used to express the win rate is to state the number of big blinds won followed by bb/100. For example, a player might say, “my win rate over my past 100,000 hands at 50NL has been 4bb/100.”.
For 6-max games on Bovada and Ignition I average 68 hands per hour. On WPN I average 77 hands per hour. In 9-max games the numbers are 55 and 62 respectively.
Why the discrepancy?
It’s because the VPIP is quite a bit higher on Bovada and Ignition. I mean it makes sense. If more people are limping and seeing flops then, of course, the games will play slower. I am not a live player but I can imagine 5/10 players enjoy a higher hands per hour than 1/2 players.What Are Your Exact Hands per Hour on Each Site?
Hold’em Manager has a handy multi-table report that shows exactly how many hands per hour are played based on how many tables are played at once. Here are my numbers:6-MaxPoker Network6-Max Hands per HourIgnition/Bovada68.47Winning Poker Network76.53
9-MaxPoker Network9-Max Hands per HourIgnition/Bovada55.36Winning Poker Network61.89
Surprisingly, the more tables I played the higher my hands per hour is. I guess this makes perfect sense since game selection pretty much goes out the window when you are mass multi-tabling.Hands per Hour Dealt Is Different Than Hands per Hour Played
If you look around the web you will see much higher numbers than I am writing about here. That’s because they are talking about hands per hour when you are actually dealt into the hand.
Raw hands per hour when actually being dealt in is like 90 per hour for 6-max and 75 per hour for 9-max. However, unless you are staying on the same table and playing for hours, lower numbers are more realistic.
What they fail to take into account that some of the time that you are sitting at a table you are not being dealt in. Hands dealt per hour does not include time players spend getting on and off the tables and sitting and waiting on big blinds. Therefore, actual hands played per hour is what you should use when trying to figure out your hourly win-rate.What About Hands Per Hour in Live Games?
Live dealt in numbers tend to be much closer to reality since there is no multi-tabling and a player will typically stay at the same table for hours. Even so, the number of hands played per hour is still much lower than online.
Hands per hour in live games can range anywhere from 20 to 30. Besides almost exclusively being 9 or 10 handed, it takes time for the dealer to shuffle and deal. That leaves little time to actually playing poker. I have a ton of respect for the patience and mental toughness of live grinders. Aggressive Table Selection Affects Hands per Hour
The longer you stay on a table the more hands per hour you are dealt. This is because you have to pull up a new table, sit down, and then wait for the big blinds. This takes time. Therefore, switching games often lowers your hands per hour.
Now I am not saying you should stop game selecting. In today’s tough poker environment, robust table selection is how money is made. One way to remedy this issue and still maintain your desired number of hands per hour is to just add one more table. If you normally two table, try three.Mass Multi-Tablers Slow down the GamesWhat Is The Best Poker Hands
I have noticed a direct correlation between the maximum number of tables allowed on a site to how many hands per hour is played. This is due to mass multi-tablers, who tend to time bank often and hold up the games.
For example, Bovada and Ignition also only allow 4 cash game tables to be played at once. WPN still allows people to play 24 tables at the same time. Get two or three guys playing 24 tables on one table and it becomes time bank city. Watch your hourly rate shrink due to atrophied hands per hour.Why Does Full Ring Play Slower Than 6-Max?
Full ring has a lower hands per hour because there are more players on the table that have to act during each hand. Also, 9-max games tend to play more passively than 6-max. Therefore, 6-max games tend to play about 20% faster than 9-max games.What If I Mix in Full Ring Games and Want to Have the Same Hands per Hour?
If you want to play the same amount of hands per hour at 9-max that you do at 6-max you will simply just need to add more tables.Rate Poker Hands
Let’s say you currently play 8 tables of 6-max and average 600 hands per hour. Since the average hands per hour at 9-max is 62, you simply divide 62 into 600 and that gives you your optimal number of full ring tables.
In this case, it is 9.68, or 10 tables.
*600/62= 9.68Fast-Fold (Zoom/Zone) Poker Hands Per Hour
In the last few years, the popularity of fast-fold poker has increased substantially due to the increase in hands per hour attainable. No longer do you have to sit back and watch a hand play out after you have folded. With the click of a button you are moved on to another hand.
This allows people to play between 200 and 300 hands per hour on each table.
What this means is that a player can use less real estate on his or her desktop and just focus on one or two tables at the time, while still achieving a substantial number of hands per hour.Drawbacks of Fast-Fold Poker
Fast-fold poker is not for the feint of heart. In fact, it takes a completely different strategy than you would play on a regular table. Maximizing hands per hour can actually be a big factor in determining how to play.
Personally, I dislike fast-fold poker not because of the strategic differences or the speed of play. Rather, I believe that the game is not really poker at all.
Observing opponents and then adjusting your strategy based on what they do is at the heart of poker. Fast-fold poker takes this away since every hand will find you in a different mix of opponents. So much for using gameflow considerations to your advantage.Final Thoughts
If you are reading this article so that you can start mass multi-tabling because you think it will increase your hourly rate substantially, I have to warn you. As hands per hour increases, ability to focus on what is going on and game select correctly tends to go out the window.
So unless you are at least a 6bb/100 winner playing one or two tables, I would not even consider trying to maximize your hands per hour yet. Your growth as a poker player is much more important than trying to increase your hourly rate.
However, once you are ready you now know how many hands per hour you can expect to play at the tables. I hope this information allows you to figure out your optimal number of tables to play to achieve your hourly rate goals. My optimal number of tables is about 6, what’s yours?Related Questions
How do I calculate hourly rate in poker?Tocalculate your hourly rate you divide your total profit by the number of hours you have played.
What is a good poker hourly rate? What can be considered a good hourly rate in poker will vary from player to player based on their goals. It will also depend on the stake being played. It is much better to focus on win-rate instead of hourly rate. A good online win-rate will range from 2-4 big blinds per 100 hands for multi-tablers and from 4 big blinds to 10 big blinds per 100 hands for single tablers. However, due to the extreme softness of some live games, live players will often enjoy a win-rate much higher.
How many hands per hour in tournament poker? Hands per hour in tournaments will be higher than cash games since there is no table selection involved. Online you can expect to play 90 hands per hour for 6-max and 75 hands per hour for 9-max. As the blinds increase, hands also tend to play faster since the game evolves into push-fold. Live tournament players play around 30 hands per hour.
This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities
Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.
___________________________________________________________________________Preliminary Calculation
Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.These are the same hand. Order is not important.
The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.
The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.
Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is
This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.
The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.
If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.
Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.
Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:
One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.
___________________________________________________________________________How To Rate Poker HandsThe Poker Hands
Here’s a ranking chart of the Poker hands.
The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.Rate Poker Hands
Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.
The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.
Definitions of Poker HandsPoker HandDefinition1Royal FlushA, K, Q, J, 10, all in the same suit2Straight FlushFive consecutive cards,all in the same suit3Four of a KindFour cards of the same rank,one card of another rank4Full HouseThree of a kind with a pair5FlushFive cards of the same suit,not in consecutive order6StraightFive consecutive cards,not of the same suit7Three of a KindThree cards of the same rank,2 cards of two other ranks8Two PairTwo cards of the same rank,two cards of another rank,one card of a third rank9One PairThree cards of the same rank,3 cards of three other ranks10High CardIf no one has any of the above hands,the player with the highest card wins
___________________________________________________________________________Counting Poker Hands
Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.
Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.
Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is
Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?
Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.
Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.
Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.
Two Pair and One Pair
These two are left as exercises.
High Card
The count is the complement that makes up 2,598,960.
The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.
Probabilities of Poker HandsPoker HandCountProbability2Straight Flush400.00001543Four of a Kind6240.00024014Full House3,7440.00144065Flush5,1080.00196546Straight10,2000.00392467Three of a Kind54,9120.02112858Two Pair123,5520.04753909One Pair1,098,2400.422569010High Card1,302,5400.5011774Total2,598,9601.0000000
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2017 – Dan Ma
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