Poker 2 Spieler Flush
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
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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.
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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.
- A flush is any five cards of the same suit. An ace-high flush is higher (and beats) a king-high flush, regardless of the other cards in each hand. When two or more players hold a flush, the hands are compared card-to-card until one hand wins (the highest next card wins, such as when A-7-6-3-2 beats A-7-5-4-3).
- Once you know that a flush beats a straight and three of a kind beats two pair, you're off to a good start. After memorizing the proper order of poker hands, you'll need to practice reading the board. And figuring out which 5-card poker hand is your best from any combination of your hole cards and the community cards.
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.
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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.
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The 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.
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 Hands
| Poker Hand | Definition | |
|---|---|---|
| 1 | Royal Flush | A, K, Q, J, 10, all in the same suit |
| 2 | Straight Flush | Five consecutive cards, |
| all in the same suit | ||
| 3 | Four of a Kind | Four cards of the same rank, |
| one card of another rank | ||
| 4 | Full House | Three of a kind with a pair |
| 5 | Flush | Five cards of the same suit, |
| not in consecutive order | ||
| 6 | Straight | Five consecutive cards, |
| not of the same suit | ||
| 7 | Three of a Kind | Three cards of the same rank, |
| 2 cards of two other ranks | ||
| 8 | Two Pair | Two cards of the same rank, |
| two cards of another rank, | ||
| one card of a third rank | ||
| 9 | One Pair | Three cards of the same rank, |
| 3 cards of three other ranks | ||
| 10 | High Card | If no one has any of the above hands, |
| the player with the highest card wins |
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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 Hands
| Poker Hand | Count | Probability | |
|---|---|---|---|
| 2 | Straight Flush | 40 | 0.0000154 |
| 3 | Four of a Kind | 624 | 0.0002401 |
| 4 | Full House | 3,744 | 0.0014406 |
| 5 | Flush | 5,108 | 0.0019654 |
| 6 | Straight | 10,200 | 0.0039246 |
| 7 | Three of a Kind | 54,912 | 0.0211285 |
| 8 | Two Pair | 123,552 | 0.0475390 |
| 9 | One Pair | 1,098,240 | 0.4225690 |
| 10 | High Card | 1,302,540 | 0.5011774 |
| Total | 2,598,960 | 1.0000000 |
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2017 – Dan Ma
The information below is for historical purposes only. Full Flush went offline in 2016. It is also important to note that they did not pay players.
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Even though Full Flush Poker went offline there are plenty of amazing sites to play at. Find our top recommended online poker site here.
Players who are on the lookout for the best possible deals in online poker might look towards some of the smaller, lesser-known poker platforms. However, this doesn’t in any way mean that players need to interact with a potentially risky platform in order to chase the most lucrative jackpots online today. Full Flush Poker is an example of this.
It is a stable, functional platform that offers excellent bonuses and rates of return without the highest possible risk. This doesn’t mean that the platform doesn’t come without warning though. Let’s take a closer look at what’s on offer. Note: Full Flush hasn’t paid its players for several months now. DO NOT SIGN-UP AT FULL FLUSH POKER.
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About Full Flush Poker
Full Flush Poker is a member of the Equity Poker Network, who themselves are owned by Equity Marketing. They hold a foreign licensed based in Curacao and has software based on Playsafe. Players shouldn’t feel like they’re out of the loop for not having heard about the platform though. There was an open beta trial for early registrants that started in September 2013 with the actual launch taking place shortly thereafter in November.
In an already saturated market, Full Flush Poker isn’t the most standout name and has yet to really offer a unique contribution that will drive players towards it. They are currently running a promotion where all deposits up to $1,500 will be matched to 150% with a 20% cash back option being available, but this hardly seems like an industry leader.
The Community
Fortunately, because Full Tilt doesn’t have the largest presence on the Internet today, the player pool is rather limited and doesn’t have a significant amount of card sharks who are waiting to take the chips away from unsuspecting beginners. It can be a great introductory platform for those who want to get to grips with online poker as a whole by playing in ring games, tournaments, and freerolls without the added risk of being devoured by a shark.
What’s more is that Full Flush openly welcomes those based in the United States, which is a particular issue for those who were affected by Black Friday or live in a state where online poker games is a questionable activity but aren’t limited to state-regulated card rooms, like in Delaware.
Full Flush Poker, on the surface, also appears to be a platform that was created specifically for beginners in mind. The majority of the poker games on offer are wagered at very low stakes that won’t make any sort of significant dent in a large bankroll. However, the game options themselves are limited to Omaha and Texas Hold’em. They’re the two most popular variants on the Internet today, but stud and other variants aren’t provided here.
Bonus Scheme
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Full Flush gives an incentive for players to deposit more than just the bare minimum ($25). When players deposit more, they’ll receive “promo bucks” that can be exchanged for freeroll tickets that entitle players to enter satellite tournaments without a buy-in that take place every Sunday and have guaranteed prize pools of at least $5,000.
Full Flush also offers “pop points” that are randomly awarded to encourage repeat play. They have a cash value of 100 points to $0.50 USD, but they’re often exchanged for tournament entries at a much better rate depending on the nature of the tournament, the prize pool, and player limits.
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Banking
Full Flush Poker initially started by only accepting credit card deposits, but gradually expanded to accept Western Union transfers. Withdrawal options are equally as limited with money orders requiring up to 10 business days after approval or a bank wire that can take up to seven business days. These times are in addition to the “know your customer” requirements that US-friendly casinos are required to abide by, which can add up to two additional weeks for security clearances and vetting procedures to be carried out.
Scandal
Unfortunately, Full Flush Poker doesn’t have the cleanest record in online poker. There have been several reports circulated regarding withdrawal requests leading to subsequent account bans due to “managerial discretion”. While players have indeed been able to cash out from the website, they are typically very low amounts and jackpots are reportedly not honored.
It can also be difficult to contact customer services. Live chat options tend to be disconnected when players reportedly become agitated at their account issues, and both telephone and snail mail options are a complete loss.
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Conclusion
It’s difficult to recommend this platform due to the scandalous reports that follow it. However, these rogue reports seem to be just a few in an otherwise acceptable mix of happy players who willingly make use of the platform for their earliest online poker ventures.
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It would be advisable to proceed with caution. There is indeed money to be won using Full Flush Poker and their popular game offers, but it comes at a risk that could otherwise be mitigated by sticking to one of the more well-known and reputable online poker providers.