Poker Royal Flush Probability
Poker Royal Flush Probability Average ratng: 6,7/10 170 votes
Originally Answered: What is the probability of getting a royal flush or straight flush in poker? There are 4 royal flushes, because that is AKQJ10 all the same suit. A royal flush is the highest straight flush, becsuse the ace is the top card. Each suit has 9 straight flushes because your low card can be 2 3 4 5 6 7 8 9 10. The probability of being dealt a royal flush is the number of royal flushes divided by the total number of poker hands. We now carry out the division and see that a royal flush is rare indeed. There is only a probability of 4/2,598,960 = 1/649,740 = 0.00015% of being dealt this hand.
- A “poker hand” consists of 5 unordered cards from a standard deck of 52. There are 52 5 = 2,598,9604 possible poker hands. Below, we calculate the probability of each of the standard kinds of poker hands. This hand consists of values 10,J,Q,K,A, all of the same suit. Since the values are fixed, we only need to choose the suit.
- Odds of Flopping the Royal Flush Straight Draw It is nearly difficult to flop this hand rank. The probability of flopping the Royal Flush straight draw is a much more probable one. Odds of Making the Royal Flush Postflop There are going to be two main types of Royal Flush draw that we will flop.
- It’s possible for the royal flush to find itself facing a lower straight flush, in which case the royal flush always wins. Royal Flush Poker Probabilities Now we’ll look at the preflop, flop, turn and river probabilities of making a Royal Flush in both Hold’em and Pot Limit Omaha.
frisbee25
What are the odds of a royal on a 100 multi-hand jacks or better game? You only get one flop initially then draw a 100 hands on this version, not sure if they are all like that.
ThatDonGuy
I have a feeling I'm making a mistake posting this without checking it through simulation first, but I get a probability of 1 in 274.564, assuming you always play a 'royal or nothing' strategy (for example, if you are dealt A-A-A-A-2, keep one of the aces and discard the other four cards; if you are dealt a suited J-10-9-8-7, keep the jack and 10; if you are dealt 9-9-9-9-6 or suited 9-8-7-6-5, discard all five cards).| High cards by suit | Number of hands |
|---|---|
| 5 | 4 |
| 4 | 4 x 5 x 32 |
| 4-1 | 4 x 5 x 15 |
| 3 | 4 x 10 x (32 x 31 / 2) |
| 3-1 | 4 x 10 x 3 x 5 x 32 |
| 3-2 | 4 x 10 x 3 x 10 |
| 3-1-1 | 4 x 10 x 3 x 25 |
| 2 | 4 x 10 x (32 x 31 x 30) / 6 |
| 2-1 | 4 x 10 x 3 x 5 x (32 x 31) / 2 |
| 2-1-1 | 4 x 10 x 3 x 25 x 32 |
| 2-1-1-1 | 4 x 10 x 125 |
| 2-2 | 6 x 100 x 32 |
| 2-2-1 | 6 x 100 x 2 x 5 |
| 1 | 4 x 5 x (32 x 31 x 30 x 29) / 24 |
| 1-1 | 6 x 25 x (32 x 31 x 30) / 6 |
| 1-1-1 | 4 x 125 x (32 x 31) / 2 |
| 1-1-1-1 | 625 x 32 |
You keep the highest number (so, for example, in 3-1, you keep the 3 cards of the same suit; in 2-2-1, you keep either of the sets of 2 cards of the same suit).
If you have five suited high cards, obviously that's a Royal Flush - in fact, 100 of them - and the probability is 1.
If you have 4, you have a 1/47 chance of getting the fifth card, which means you have a 46/47 chance of not getting it, so you have a (46/47)100 chance of not getting any royals in 100 of these hands, and a 1 - (46/47)100 chance of getting at least one.
Poker Royal Flush Probability Calculator
Similarly, with 3 suited high cards, you have a 1 - (1 - 2 / (46 x 47) )100
Poker Royal Flush Probability Charts
chance of at least one royal;
with 2, a 1 - (1 - 6 / (47 x 46 x 45) )100 chance;with 1, a 1 - (1 - 24 / (47 x 46 x 45 x 44) )100
Poker Royal Flush Probability Table
chance;with none, a 1 - (1 - 120 / (47 x 46 x 45 x 44 x 43) )
100 chance.Multiply the numbers of each hand by the appropriate probability, add them together, and divide by 2598960.
frisbee25
Has anyone claimed to beat any of these multi hand video poker games or just single hand? Is variance increasing for these multi-hand games because it's not a 100 unique flops or is it the same?
Probabilitiesfor 5 card poker hands with misc. wild cards
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.)
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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.)
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