Poker Probability Of Flush
Introduction
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. The probability of getting a flush is the ratio of the number of ways of getting a flush divided by the total number of hands; it is 51 = 33/16660 =.17. Not very high odds - about 2 in every 1000 hands! 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. Three of a Kind.
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.
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 |
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
Poker Probability Royal Flush
| 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.
Written by: Michael Shackleford
Getting a royal flush is the hardest hand to obtain when playing poker online or in a casino. If you’re wondering what your odds are of being dealt a royal flush and other hands, you’ve come to the right place. We’ve developed this page to equip you with all the information you need to know about your poker hand odds.
In this detailed guide about your odds of being dealt a royal flush and other hands while playing poker, we’ll provide you with tons of information. You can check out the preview below to get an idea of everything we’ll cover. Feel free to click on one of these section titles if you want to jump ahead.
Breakdown of Potential Poker Hands
Before we dive into royal flush odds and other hands, we wanted to first ensure you’ve got a good understanding of the different hands possible when playing poker. Check out the sections below to look over all the different poker hands. We’ve listed them in the order of their rank when playing the game.
No Pair
This one should be pretty obvious. In casino poker and online poker, if you don’t have a single pair or higher in your hand, you have what’s considered a “no pair” hand. In this case, your hand’s value will depend on the highest card you’ve got.
Single Pair
If you end up getting a one pair hand, it means you’ve got two card values that match in your hand. For example, if you have two 4s, you have a single pair of 4s. While this isn’t a powerful poker hand, it does outrank anyone who has a no pair hand.
Two Pair
Kicking things up a slight notch from a single pair would be a two pair hand. In this scenario, you have two sets of matching card values. As an example, if you have two Ks and two 10s in your hand, it would be a two pair hand. In turn, it would outrank any players with just a single pair or no pair.
Three of a Kind
As the name implies, a poker hand that counts as three of a kind has three cards of the same value. For example, if you have three jacks in your hand, this would create a three of a kind poker hand. If you end up with the three of a kind hand, you’ll have a better hand than no pair, single pair, and two pair hands.
Straight (Not Royal or Flush)
Up next on the poker hand rank scale is a straight. Here, we’re only focused on standard straights, which means we’re not counting straights that are either flush or royal in nature (more on those in a moment). To make a straight, you’ll need all five cards in your hand to be in sequential order. As an example, if you had A, 2, 3, 4, and 5, you’d have a straight poker hand.
Flush (Not Straight or Royal)
Topping out straights and the other hands below it, a flush is another form of a poker hand. With a flush, you’ll have all five cards of your poker hand of the same suit. As an example, if all five cards in your hand are spades, you have a flush. For this particular hand, your cards do not count as a straight flush or a royal flush. We’ll touch on each of those below.
Full House
The next hand up the poker hand ranking scale is a full house. To make a full house with your hand, you’ll need to have a three of a kind paired with a two of a kind. If you have three 10’s and two 5’s, you’d have a full house.
Four of a Kind
One of the toughest hands to get when playing poker is a four of a kind. Here, you’ll need to have four cards of the same value in your hand. As an example, if you had four queens in your hand, you’ll have made a four of a kind poker hand. With four of a kind, there are only two other poker hands that can beat you.
Straight Flush (Not Royal)
Probability Of A Flush Poker
Second from the top of the best poker hands possible is the straight flush. The flush portion of this name implies you’ll need all your cards to be of the same suit. However, to make a straight flush, they also must be in sequential order. For example, having 3, 4, 5, 6, and 7 of the same suit would provide you with a straight flush poker hand.
Royal Flush
The king of all poker hands is the royal flush. With a royal flush, it’s essentially a very specific straight flush. For starters, all your five cards must be the same suit. On top of that, it must be the 10, J, Q, K, and A of a particular suit to complete the royal flush.
Poker Hand Odds for Five-Card Games
Up first, we wanted to start by presenting you with your odds of being dealt a royal flush and other hands when playing five-card games of poker. Most notably, this will include Five-Card Stud Poker. We’ve included a chart below which showcases your odds of being dealt each hand in conjunction with the potential combinations and associated probability.
One thing worth noting is that the chart below showcases your odds of having one of the hands in a five-card poker game. This data does not account for any possibilities of wild cards or draws, which may be present in select games like Five-Card Draw.
| Poker Hand | Odds | Combinations | Probablity |
|---|---|---|---|
| Royal Flush | 1 in 649,740 | 4 | 0.00015% |
| Straight Flush | 1 in 72,192 | 36 | 0.00139% |
| Four of a Kind | 1 in 4,165 | 624 | 0.02401% |
| Full House | 1 in 693 | 3,744 | 0.14406% |
| Flush | 1 in 508 | 5,108 | 0.19654% |
| Straight | 1 in 254 | 10,200 | 0.39246% |
| Three of a Kind | 1 in 46.2 | 54,912 | 2.11285% |
| Two Pair | 1 in 21 | 123,552 | 4.75390% |
| Single Pair | 1 in 1.37 | 1,098,240 | 42.25690% |
| No Pair | 1 in 0.995 | 1,302,540 | 50.11774% |
Chart Labels
- Odds: The odds of being dealt the particular poker hand in a five-card game.
- Combinations: How many different ways the poker hand can be made using all 52 cards in the deck.
- Probability: The statistical probability of being dealt the hand in a five-card poker game.
As you can see from the chart above, you’ve got the highest chance of being dealt a no pair or single pair hand when playing a five-card variant of poker online or in a casino. Interestingly, there’s roughly a 50% chance you won’t have a pair or better.
However, you can see just how tough it can be to get some of the other higher-ranking poker hands. Even two pair hands only happen about 5% of the time. And if you’re hoping for a royal flush, the odds of it happening are minuscule.
Things More Likely to Happen Than Being Dealt a Royal Flush
Since the royal flush is the hardest poker hand to achieve, we wanted to provide you with some visualizations to help you grasp just how rare it is. Check out the list of things below, which are more likely to happen to you than being dealt a royal flush when playing a five-card variant of poker.
Getting in a Car Accident
1 in 103
Getting Audited by the
Internal Revenue Service (IRS)
1 in 175
Winning an Academy Award
1 in 11,500
Losing an Appendage
in a Chainsaw-Related Accident
1 in 4,464
Going to the ER
With a Pogo Stick-Related Injury
1 in 103
Poker Hand Odds for Seven-Card Games
Up next, we wanted to provide you with royal flush odds and other poker hands when playing seven-card versions of poker. If you’re into games like Seven-Card Stud and No Limit Texas Hold’em, this is the section for you.
While the addition of two extra cards to work with doesn’t sound like much to some, it creates a dramatic difference. Instead of just 2,598,960 potential hand combinations, playing poker with seven cards brings the possibility of 133,784,560 hands. That means there are more than 50 times as many possible hand combinations thanks to those extra two cards in play!
This chart focuses on your odds of being dealt one of these hands in a game of seven-card poker. As with the previous five-card section, the poker probability and odds below do not take into account wild cards and draws from specific versions of poker.
| Poker Hand | Odds | Combinations | Probablity |
|---|---|---|---|
| Royal Flush | 1 in 30,939 | 4,324 | 0.00323% |
| Straight Flush | 1 in 3,589 | 37,260 | 0.02785% |
| Four of a Kind | 1 in 594 | 224,848 | 0.16807% |
| Full House | 1 in 37.5 | 3,473,183 | 2.59610% |
| Flush | 1 in 32.1 | 4,047,644 | 3.02549% |
| Straight | 1 in 20.6 | 6,180,020 | 4.82987% |
| Three of a Kind | 1 in 19.7 | 6,461,620 | 23.49554% |
| Two Pair | 1 in 3.26 | 31,433,400 | 23.49554% |
| Single Pair | 1 in 1.28 | 58,627,800 | 43.82255% |
| No Pair | 1 in 4.74 | 23,294,460 | 17.41192% |
Chart Labels
- Odds: The odds of being dealt the particular poker hand in a seven-card game.
- Combinations: How many different ways the poker hand can be made using all 52 cards in the deck.
- Probability: The statistical probability of being dealt the hand in a seven-card poker game.
Immediately, you’ll probably notice how much better your odds of getting most hands are. In the next section, we’ll provide you with even more information about how much better your chances are for each of these hands if you play a seven-card variant instead of a five-card one.
Thanks to the additional two cards, offering you the chance to make your best five-card hand, there are more potential combinations which can help you improve your starting hand.
How Much Better Your Odds Are Playing Seven-Card Poker
Now that we’ve broken down the difference in royal flush odds and other poker hands between five- and seven-card poker games, we wanted to help you visualize just how much better your odds are when playing a seven-card game. Check out the chart below to see why you might opt to choose a seven-card game if you’re hoping to land a significant hand like a royal or straight flush.
| Poker Hand | Percentage Increase |
|---|---|
| Royal Flush | 2000.00% |
| Straight Flush | 1910.64% |
| Four of a Kind | 600.00% |
| Full House | 1702.13% |
| Flush | 1439.38% |
| Straight | 1077.02% |
| Three of a Kind | 128.60% |
| Two Pair | 394.24% |
| Single Pair | 3.71% |
| No Pair | -65.26% |
Probability Of Straight Flush Poker
As you can see from the chart above, there’s a 2000% greater chance you’ll get a royal flush when playing a seven-card poker game instead of a five-card game. Other hands which have an increased chance of happening when you’re playing a seven-card variant of poker include the straight flush, full house, flush, and straight.
Interestingly, there’s one hand where you have a lower chance of getting it when playing a seven-card game of poker instead of a five-card game. That hand is the no pair hand. Intuitively, this makes sense since there are increased chances you’ll make at least a pair thanks to the expanded cards you’re playing with. In this case, your chance of getting a no pair hand is 65% less when playing a seven-card game as opposed to a five-card one.
Wrap Up
Thanks for stopping in to check out this page about poker probability and the odds of being dealt a royal flush when playing online poker and casino poker. If you’re planning to play poker soon, don’t miss our complete guide to real money poker. In it, you’ll find all sorts of helpful information, including terminology, strategies, and so much more.
If you enjoyed this page about the odds of getting a royal flush, you might also enjoy other pages we’ve developed in this series. Check out the choices below to explore some of our other “What Are the Odds?” pages.