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
Step 1: Determine the probability of 0 to 6+ royals in the first 5,000 hands. Let's assume the probability of a royal is. Step 2: Consider there to be seven states for the remaining 24,995,000 hands. For each one, the previous 5,000 hands can. Step 3: Develop the transition matrix for the odds. 0.00139% Chance of Being Dealt a Straight Flush Q: What are the Odds of Hitting a Straight Flush in Hold'em? A: The 'straight flush' is the second strongest hand in Hold'em (or Omaha), ranking behind only the 'Royal Flush'. The 'straight flush. Your odds being dealt a natural Royal Flush with the first 5 cards out of the deck when playing the traditional game are just 1 in 649,740. To put that number into some perspective, if you were dealt 20 poker hands every night over the course of your lifetime you would expect to get a royal flush.
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.
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.
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.
Video Poker Royal Flush Odds
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.
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
|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|
Counting Poker Hands
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.
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?
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.
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.
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
|3||Four of a Kind||624||0.0002401|
|7||Three of a Kind||54,912||0.0211285|
2017 – Dan Ma
The ins and outs of that most elusive of hands
By Henry Tamburin
I get many questions about a royal flush in video poker. That’s not too surprising since the royal flush is the premier hand that all video poker players dream (and hope) of getting. Here’s a sample of questions and my responses.
Q: I’ve been playing video poker several times a week for over a year. You keep saying that a royal flush occurs once in every 40,000 hands yet I still haven’t gotten a royal. What gives?
Firstly, I never wrote that you could expect one royal flush after playing 40,000 hands (or one cycle). What I wrote was, “On average, you will hit a royal flush once in every 40,000 hands.” The word “average” means a whole bunch of sets of 40,000 hands. In other words, in any given set of 40,000 hands, you could hit more than one royal flush or, heaven forbid, possibly no royals. In fact, you have a 36.8% chance that you won’t get a royal in one cycle (40,000 hands), and a 13.5% chance after two cycles (80,000 hands). Ouch! Therefore, the fact that you went over a year without a royal is statistically possible.
Q: How come every time I need one card for a royal flush, it never shows up, but that exact card that I needed always seems to show up on the very next hand?
That’s because you have “selective memory.” The computer program in the video poker machine that randomly selects the cards for each hand doesn’t use the information from previous hands to determine which cards it will deal. Every hand is a random deal regardless of what cards appeared (or didn’t appear) on the previous hand.
Q: Over three years, I hit seven royal flushes in the same casino and none in two other casinos that I play regularly. I’m beginning to believe those casinos somehow tighten their video poker machines so players can’t get a royal.
You will average one royal flush per roughly every 40,000 hands at any casino. Casinos can’t change the odds of hitting a royal flush. (What they can do is change the payout … some casinos will pay less than 4,000 coins for a royal flush; therefore, always check to be sure that the payout for a five-coin royal flush is 4000 coins.) The bottom line is as long as the pay schedule is the same for a particular video poker game, the odds of getting a royal flush will be the same no matter where the machine is located (assuming a random deal).
Q: I’ve been dealt many three- and four-card royal flushes lately. What are the odds of this happening?
Best Video Poker Odds
Playing Jacks or Better, you’ll experience the thrill of being dealt a four-card royal flush once in every 2,777 hands (roughly once every four hours on average). Once in every 92 hands, on average, you’ll be dealt a three-card royal flush (about 7-8 per hour). This is what makes video poker exciting; namely, that you’ll have several opportunities to draw for a royal flush even if the odds are somewhat long (see next question).
Q: When you hold three cards to the royal flush, what is the chance of getting the two cards that you need on the draw for a royal flush?
You have a one in 1,081 chance of getting the two cards you need for the royal flush. The following table shows the chance of hitting the royal flush on the draw when you hold x cards to the royal flush.
|RF Cards in Initial Five-Card Hand||Chance of Hitting the Royal Flush |
|1 in 383,484|
|1 in 178,365|
|1 in 16,215|
|1 in 1,081|
|1 in 47|
Q: If I’m dealt a three-card royal flush and a high pair in the same hand, why does the strategy say to hold the high pair when the royal flush pays so much more?
You need to analyze all the possible winning hands that you could get when you hold a three-card royal flush vs. when you hold a high pair in the same hand. These calculations have already been done for you. For example, suppose your initial hand contains 10-J-Q of diamonds along with a queen of clubs. The expected return (ER) for holding the pair of queens is 7.6827 vs. 7.4098 for holding the three-card royal flush (this is for 9/6 Jacks or Better). In dollars and cents, you’d earn 27 cents more on average for a max coin wager on a dollar denomination machine by holding the high pair vs. the three-card royal flush in this example.
Q: My wife plays Jacks or Better. The other day she was a dealt a hand that contained a four-card straight flush with a gap and a three-card royal flush. She held the three-card royal flush. Was that the correct play?
I’m sorry to say it wasn’t. The correct play was to hold the four-card straight flush—even with a gap—over the three-card royal flush. (Tip: If your wife had a strategy card with her, she would have made the right play.)
Q: What are the odds of being dealt a royal flush in the initial hand?
The odds are one in 649,740 hands. You might think that’s close to impossible but it could happen. (This happened to me once while I was showing my father-in-law how to play a Triple Line video poker game in a Las Vegas casino, resulting in a royal flush on each line. How’s that for luck?)
Video Poker Royal Flush Odds
Q: How much does the royal flush contribute to the 99.54% return for 9/6 Jacks or Better?
The royal flush contributes 1.9807% toward the overall 99.64% return. The following table summarizes the contribution of each winning hand toward the overall 99.54% return (for 9/6 Jacks or Better). When you don’t hit the royal or straight flush, the best return you can expect, even playing perfectly, is about 97%.
|Hand||Contribution to Return|
|Four of a Kind||5.9064%|
|Three of a Kind||22.3346%|
Got a video poker question? Send it to [email protected]
Tamburin’s Tip of the Month
You are playing NSU Deuces Wild. How would you play these hands that don’t contain a deuce?
In the top hand, your best play is to hold the consecutive three-card straight flush 6-7-8 (2.77 ER) over the four card straight 5-6-7-8 (2.55 ER). In the bottom hand, because the three-card straight flush has a gap (2.47 ER) your best play is to hold the consecutive four-card straight 4-5-6-7. When you play NSU Deuces Wild and your initial hand doesn’t contain a deuce, you should hold a consecutive three-card straight flush (5-6-7 through 9-10-J) over a consecutive four-card straight (from 4-5-6-7 to 10-J-Q-K), but the latter over a three-card straight flush with one or two gaps.
Royal Flush Odds Texas Holdem
Royal Straight Flush In Poker
Henry Tamburin is a blackjack and video poker expert. He is the host of the smartgaming.com website and the editor of the Blackjack Insider newsletter (for a free three-month subscription, visit www.bjinsider.com/freetrial). For a free copy of his Casino Gambling Catalog, which contains books, strategy cards, and software for video poker players, call toll free 1-888-353-3234, or visit the web store at smartgaming.com.