Poker Probability Of Flush
Bottom line: In stud poker, the probability of an ordinary flush is 0.0019654. On average, it occurs once every 509 deals. I just ran the numbers and thought it was worth posting. Using Combination. Flops combin(50, 3) = 19600. 3 flush combin(11, 3) = 165 / 19600 = 0. = 0.84% = 1/119.
Probabilitiesfor 6 card poker hands with misc. wild cards
Probabilitiesfor 7 card poker hands with misc. wild cards
Probabilitiesfor 8 card, 9 card, and 10 card poker hands with misc. wild cards
Lowball (Low Ball) poker probabilities with misc. wild cards (5 to 10 cards)
http://www.durangobill.com/LowballPoker/Lowball_Poker.html
Click here for optimal strategy and expected value for Video Poker
http://www.durangobill.com/VideoPoker.html
The probability of being dealt various poker hands has been printed in many other sources. We present the probabilities for a 5 card deal here, and then concentrate on how to calculate these numbers.
Poker Hand Number of Combinations Probability
--------------------------------------------------------
Royal Straight Flush 4 .0000015391
Other Straight Flush 36 .0000138517
Four of a kind 624 .0002400960
Full House 3,744 .0014405762
Flush 5,108 .0019654015
Straight 10,200 .0039246468
Three of a kind 54,912 .0211284514
Two Pairs 123,552 .0475390156
One Pair 1,098,240 .4225690276
High card only 1,302,540 .5011773940
Total 2,598,960 1.0000000000
(See Probabilitiesfor 5 card poker hands with misc. wild cards for additional details.)
The first calculation that must be made is to determine the total possible poker hands. A poker hand consists of 5 cards randomly drawn from a deck of 52 cards. Thus, the number of combinations is COMBIN(52, 5) = 2,598,960. Each of these 2,598,960 hands is equally likely. For each of the above “Number of Combinations”, we divide by this number to get the probability of being dealt any particular hand.
For the calculations, we will first split out the “No Pair” hands which include Royal Straight Flushes, Straight Flushes, Flushes, Straights, and “Nothings”. Then, we will look at all combinations that have at least 1 pair.
The cards in a hand without any pairs will have 5 different denominations selected randomly from the 13 available (2, 3, 4...Ace). Also, each of the 5 denominations will select 1 suit from the four available suits. Thus the total number of no-pair hands will equal:
COMBIN(13, 5) * (COMBIN(4, 1))^5 = 1287 * 1024 = 1,317,888.
A Straight Flush consists of 5 consecutive cards in the same suit and may have a high card of 5, 6, 7, 8, 9, 10, Jack, Queen, King, or Ace for a total of 10 different ranks. Each of these may be in any of 4 suits. Thus there are 40 possible Straight Flushes. An Ace high Straight Flush is a Royal Flush. Since there are only 4 different suits, there are only 4 possible Royal Straight Flushes. When we subtract the 4 Royal Straight Flushes from the total of 40 Straight Flushes, we are left with 36 other Straight Flushes that are King high or less.
A Flush consists of any 5 of the 13 cards from a particular suit. There are 4 possible suits. Thus the number of possible Flushes is: COMBIN(13, 5) * 4 = 5,148. However, this includes the 40 possible Straight Flushes. When we subtract these out, we are left with: 5,148 - 40 = 5,108 possible ordinary Flushes.
A Straight consists of 5 cards with consecutive denominations and may have a high card of 5, 6, 7, 8, 9, 10, Jack, Queen, King, or Ace for a total of 10 different ranks. Each of these 5 cards may be in any of the 4 suits. Thus there are 10 * 4^5 = 10,240 different possible straights . However, this total includes the 40 possible Straight Flushes. Thus we subtract 40, which leaves us with 10,200 possible ordinary Straights.
Finally, we come to the “Nothing” hands which are basically all the left over garbage. This is simply the total number of “No Pair” hands minus all the good stuff. This gives us: 1,317,888 - 4 - 36 -5,108 - 10,200 = 1,302,540 “Nothing” hands.
Now on to 1 pair or better. A hand with just 1 pair has 4 different denominations selected randomly from the 13 available denominations. 3 of these denominations will select 1 card randomly from the 4 available suits. The 4th denomination will select 2 cards from the available 4 suits. Finally, the pair can be any one of the four available denominations. Thus the calculation is: COMBIN(13, 4) * (COMBIN(4, 1))^3 * COMBIN( 4, 2) * 4 = 1,098,240 possible hands that have just one pair.
The calculation for a hand with two pairs is similar. We will have 3 random denominations taken from the 13 available. Two of these denominations will use 2 of the four available suits while the third denomination selects 1 of the four available suits. The singleton card may be any one of the three denominations. Thus, the calculation becomes: COMBIN(13, 3) * (COMBIN(4, 2))^2 * COMBIN(4, 1) * 3 = 123,552 possible hands with 2 pairs.
Three of a kind is calculated in a similar manner. There will be 3 different denominations from the 13 possible denominations. One denomination will select 3 of the 4 available suits while the other two denominations select 1 card from each of the 4 possible suits. Finally, the three of a kind can be in any of the three denominations. The calculation becomes: COMBIN(13, 3) * COMBIN(4, 3) * (COMBIN(4, 1))^2 * 3 = 54,912 possible hands with 3 of a kind.
The next calculation will be for a Full House. A Full House only uses 2 of the 13 denominations. One of these will select 3 cards from the 4 available while the other selects 2 cards from the 4 available. Finally the denomination that has 3 cards can be either one of the 2 denominations that we are using. This gives us: COMBIN(13, 2) * COMBIN(4, 3) * COMBIN(4 , 2) * 2 = 3,744 possible Full Houses.
The final calculation is for 4 of a kind. Again, we will select 2 denominations from the 13 available. One of these will select 4 cards from the 4 available (Obviously the only way to do this is to take all four cards.) while the other denomination takes 1 of the available 4 cards. The denomination that has 4 of a kind can be either one of the 2 available denominations. Thus, the calculation becomes: COMBIN(13, 2) * COMBIN( 4, 4) * COMBIN( 4, 1) * 2 = 624 different ways of being dealt 4 of a kind. (On the draw, ask one of the other players what the odds are of drawing to an inside straight. Then draw your card. It won't make any difference though as no one else will have anything, and they will all fold.)
Return to Durango Bill's Home page
Web page generated via Sea Monkey's Composer HTML editor
within a Linux Cinnamon Mint 18 operating system.
(Goodbye Microsoft)
- 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.
- Flush or 4 flush = 0. = 12% = 1 / 8.5 4 flush hit on the turn = 9 / 47 = 0. = 19% = 1 / 5.22 4 flush hit on the river = 8 / 46 = 0. = 20% = 1 / 5.11 flush hit on turn or river or both.
- It's worth mentioning that there is an additional (19.4%. 17.4%) = 3.33% chance of completing the flush on the turn and seeing another flush card on the river. Because players going all-in for a flush draw after the flop usually have near the nuts, this 3.33% outcome means the pot odds calculation depends on how high your flush is.
POKER PROBABILITIES
- Texas Hold'em Poker
Texas Hold'em Poker probabilities - Omaha Poker
Omaha Poker probabilities - 5 Card Poker
5 Card Poker probabilities
POKER CALCULATOR
- Poker calculator
Poker odds calculator
POKER INFORMATION
- 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.
| 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
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:
Poker Probability Of Flush Valve
- 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 Flush Rules
- 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:
This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.
Home > 5 Card Poker probabilities