The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739: 1.
Introduction
High Card Flush made its debut at Harrah's Laughlin in summer 2011. In February 2013 it found another placement at the M in Las Vegas. After that slow beginning the game caught on and today has lots of placements.
Poker Hand Number of Combinations Probability Royal Straight Flush 4. Other Straight Flush 36. The Odds of Making a Flush Hand in Poker The odds of flopping a Flush with a suited starting hand is 0.82% or 1 in 122 Definition of Flush – We make a Flush by having five cards of the same suit. Bottom line: In stud poker, the probability of an ordinary flush is 0.0019654. On average, it occurs once every 509 deals. Calculate probability of a flush in poker. Ask Question Asked 1 year, 1 month ago. Active 1 year ago. Viewed 388 times 0. I have the code to keep going through a.
The game follows a fold or raise structure, like Caribbean Stud Poker and Three Card Poker. Where it differs is in the hand ranking, which is all about making the highest possible flush out of seven cards.
Rules
- High Card Flush is played with a standard 52-card deck of playing cards.
- To begin play, each player makes the mandatory Ante wager, and if desired, the optional Bonus wager.
- The player and dealer each receive seven cards face down.
- Hands are evaluated in the following fashion:
- The first ranking criteria is the greatest number of cards in any one suit. This is referred to as the 'maximum flush.' For instance, any hand with a maximum four-card flush beats any hand with a maximum three-card flush, but loses to any hand with a maximum five-card flush.
- The second ranking criteria is the standard poker-rankings for flushes; that is, a hand with a maximum four-card flush of K-Q-J-T would beat a hand with a maximum four-card flush of K-Q-J-9, but lose to a hand with a maximum four-card flush of A-4-3-2.
- Each player then decides upon one of the following options:
- Fold, and surrender the Ante.
- Raise, placing a second bet equal to at least the Ante. The maximum amount of the Raise wager depends on the rank of the player?s hand:
- With a two-, three- or four-card flush, the maximum Raise wager is equal to the Ante wager.
- With a five-card flush, the maximum Raise wager is double the Ante wager.
- With a six- or seven-card flush, the maximum Raise wager is triple the Ante wager.
- Once all players have decided, the dealer turns over his seven cards and evaluates his hand in a similar fashion as described above.
- If the dealer does not have at least a three-card flush, nine-high, all remaining players have their Antes paid, and the Raise bets are pushed.
- If the dealer has at least a three-card flush, nine-high, his hand is compared to each other player:
- All players with a higher-ranking hand win, and have their Ante and Raise wagers paid at even money.
- All players with a lower-ranking hand lose, and have their Ante and Raise wagers collected.
- Players with the exact same ranking hand as the dealer push both their Ante and Raise wagers.
- Finally, any player who made the Bonus wager has his hand evaluated against the Bonus paytable, and the Bonus wager is either paid or collected as necessary.
Mousseau Strategy
Charles Mousseau determined that without regard to cards not part of the highest flush, a close to perfect strategy is to raise on T-8-6 or higher. The player should always make the largest allowed Raise bet. This strategy has a house edge of 0.06% higher than optimal strategy.
That means to raise any four-card or higher flush, and any three-card flush of rank T-8-6 or greater. For example, you would raise J-3-2, but fold T-7-5.
The following table shows the probability and return for each possible event under the Mousseau strategy. The lower right cell shows a house edge of 2.71%.
Mousseau Strategy Return Table
| Event | Pays | Probability | Return |
|---|---|---|---|
| Player raises 3x, dealer qualifies, player wins | 4 | 0.001604 | 0.006416 |
| Player raises 2x, dealer qualifies, player wins | 3 | 0.021374 | 0.064121 |
| Player raises 1x, dealer qualifies, player wins | 2 | 0.258352 | 0.516703 |
| Player raises 1x, dealer does not qualify | 1 | 0.160076 | 0.160076 |
| Player raises 2x, dealer does not qualify | 1 | 0.006590 | 0.006590 |
| Player raises 3x, dealer does not qualify | 1 | 0.000444 | 0.000444 |
| Player raises 1x, dealer qualifies, player pushes | 0 | 0.000839 | 0.000000 |
| Player raises 2x, dealer qualifies, player pushes | 0 | 0.000001 | 0.000000 |
| Player raises 3x, dealer qualifies, player pushes | 0 | 0.000000 | 0.000000 |
| Player folds | -1 | 0.320589 | -0.320589 |
| Player raises 1x, dealer qualifies, player loses | -2 | 0.229568 | -0.459136 |
| Player raises 2x, dealer qualifies, player loses | -3 | 0.000559 | -0.001678 |
| Player raises 3x, dealer qualifies, player loses | -4 | 0.000003 | -0.000013 |
| Totals | 1.000000 | -0.027065 |
Under the Mousseau strategy, the average final wager is 1.712 units. Thus, the element of risk is 2.706%/1.712 = 1.581%.
High Card Flush Advanced Strategy
Wizard of Odds contributor Gordon Michaels has published a High Card Flush Advanced Strategy. His strategy considers the suit distribution of the penalty cards with T-3-2 to T-9-8. The bottom line is a house edge of 2.6855%. Please click the link for the specifics.
Optimal Strategy
An optimal strategy has yet to be put in writing. However, we can narrow it down, as follows.
- Make maximum raise bet with J-9-6 or higher.
- Fold 9-7-4 or lower.
- You're on your own with 9-7-5 to J-9-5.
The following table shows that under the unknown optimal strategy the house edge is 2.64%.
Optimal Strategy Return Table
| Event | Pays | Probability | Return |
|---|---|---|---|
| Player raises 3x, dealer qualifies, player wins | 4 | 0.001618 | 0.006473 |
| Player raises 2x, dealer qualifies, player wins | 3 | 0.021472 | 0.064417 |
| Player raises 1x, dealer qualifies, player wins | 2 | 0.258181 | 0.516361 |
| Player raises 1x, dealer does not qualify | 1 | 0.160038 | 0.160038 |
| Player raises 2x, dealer does not qualify | 1 | 0.006617 | 0.006617 |
| Player raises 3x, dealer does not qualify | 1 | 0.000448 | 0.000448 |
| Player raises 1x, dealer qualifies, player pushes | 0 | 0.000840 | 0.000000 |
| Player raises 2x, dealer qualifies, player pushes | 0 | 0.000001 | 0.000000 |
| Player raises 3x, dealer qualifies, player pushes | 0 | 0.000000 | 0.000000 |
| Player folds | -1 | 0.321365 | -0.321365 |
| Player raises 1x, dealer qualifies, player loses | -2 | 0.228857 | -0.457715 |
| Player raises 2x, dealer qualifies, player loses | -3 | 0.000560 | -0.001679 |
| Player raises 3x, dealer qualifies, player loses | -4 | 0.000003 | -0.000013 |
| Totals | 1.000000 | -0.026418 |
Under the Mousseau strategy, the average final wager is 1.711 units. Thus, the element of risk is 2.642%/1.711 = 1.544%.
Miscellaneous statistics:
- All told, when the player plays optimally, the player will raise 67.86% of the time.
- The dealer will have a qualifying hand 75.36% of the time.
- The player and dealer will tie 0.08% of the time.
- The standard deviation is 1.63.
Flush Bet
I have heard of two pay tables for the Flush bet. The following three tables show the details.
Pay Table 1
| Cards | Pays | Probability | Return | |
|---|---|---|---|---|
| 7 | 300 | 6,864 | 0.000051 | 0.015392 |
| 6 | 100 | 267,696 | 0.002001 | 0.200095 |
| 5 | 10 | 3,814,668 | 0.028514 | 0.285135 |
| 4 | 1 | 26,137,540 | 0.195370 | 0.195370 |
| 3 or less | -1 | 103,557,792 | 0.774064 | -0.774064 |
| Total | 133,784,560 | 1.000000 | -0.078072 |
Poker Probability Of Flush Rules
Pay Table 2
| Cards | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| 7 | 300 | 6,864 | 0.000051 | 0.015392 |
| 6 | 75 | 267,696 | 0.002001 | 0.150071 |
| 5 | 5 | 3,814,668 | 0.028514 | 0.142568 |
| 4 | 2 | 26,137,540 | 0.195370 | 0.390741 |
| 3 or less | -1 | 103,557,792 | 0.774064 | -0.774064 |
| Total | 133,784,560 | 1.000000 | -0.075292 |
Straight Flush Bet
The Straight Flush side bet pays according to the longest straight flush the player can make. I observed it only at the Planet Hollywood. The lower right cell shows a house edge of 13.11%.
Straight Flush Side Wager
| Cards | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| 7 | 8000 | 32 | 0.000000 | 0.001914 |
| 6 | 1000 | 1,592 | 0.000012 | 0.011900 |
| 5 | 100 | 39,960 | 0.000299 | 0.029869 |
| 4 | 60 | 676,196 | 0.005054 | 0.303262 |
| 3 | 7 | 8,642,932 | 0.064603 | 0.452224 |
| 2 or less | -1 | 124,423,848 | 0.930031 | -0.930031 |
| Total | 133,784,560 | 1.000000 | -0.130864 |
Internal Links
Acknowledgements
- Thanks for Charles Mousseau for providing the math for this game, except on the Straight Flush side bet. Charles' web site is tgscience.com.
- Gordon Michaels for his High Card Flush Advanced Strategy.
Poker Probability Of Flush Valve
Written by: Michael ShacklefordPOKER PROBABILITIES
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- Poker hand rankings
Ranking of poker hands
In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.
Frequency of 5-card poker hands
The following enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands). The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)
The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields as above.
Poker Probability Of Flush Toilets
| Hand | Frequency | Approx. Probability | Approx. Cumulative | Approx. Odds | Mathematical expression of absolute frequency |
|---|---|---|---|---|---|
| Royal flush | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
| Straight flush (excluding royal flush) | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
| Four of a kind | 624 | 0.0240% | 0.0256% | 4,164 : 1 | |
| Full house | 3,744 | 0.144% | 0.170% | 693.2 : 1 | |
| Flush (excluding royal flush and straight flush) | 5,108 | 0.197% | 0.367% | 507.8 : 1 | |
| Straight (excluding royal flush and straight flush) | 10,200 | 0.392% | 0.76% | 253.8 : 1 | |
| Three of a kind | 54,912 | 2.11% | 2.87% | 46.3 : 1 | |
| Two pair | 123,552 | 4.75% | 7.62% | 20.03 : 1 | |
| One pair | 1,098,240 | 42.3% | 49.9% | 1.36 : 1 | |
| No pair / High card | 1,302,540 | 50.1% | 100% | .995 : 1 | |
| Total | 2,598,960 | 100% | 100% | 1 : 1 |
The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.
When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.
Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.
The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.
Derivation of frequencies of 5-card poker hands
Poker Probability Of Being Dealt A Flush
of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).
- Straight flush — Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (10-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is:
- Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is
- or simply . Note: this means that the total number of non-Royal straight flushes is 36.
- Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is
- Four of a kind — Any one of the thirteen ranks can form the four of a kind by selecting all four of the suits in that rank. The final card can have any one of the twelve remaining ranks, and any suit. Thus, the total number of four-of-a-kinds is:
- Full house — The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and consists of three of the four suits. The pair can be any one of the remaining twelve ranks, and consists of two of the four suits. Thus, the total number of full houses is:
- Flush — The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is:
- Straight — The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-10. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving:
- Three of a kind — Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The remaining two cards can have any two of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of three-of-a-kinds is:
- Two pair — The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is:
- Pair — The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is:
- No pair — A no-pair hand contains five of the thirteen ranks, discounting the ten possible straights, and each card can have any of the four suits, discounting the four possible flushes. Alternatively, a no-pair hand is any hand that does not fall into one of the above categories; that is, any way to choose five out of 52 cards, discounting all of the above hands. Thus, the total number of no-pair hands is:
Poker Probability Of Full House
- Any five card poker hand — The total number of five card hands that can be drawn from a deck of cards is found using a combination selecting five cards, in any order where n refers to the number of items that can be selected and r to the sample size; the '!' is the factorial operator:
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