Texas Holdem Royal Flush Probability

7/18/2022

Texas Holdem Royal Flush Probability Rules
Ultimate Texas Holdem Royal Flush Odds
Probability Of A Royal Flush In Texas Holdem

There are very few feelings in poker that beat the one of flopping quads or a straight flush. There you are, sitting with the absolute nuts and your only job is to figure out how to milk your opponents most money you can.

But if you’ve been playing poker for a little while, you’ve probably realized these rare hands don’t come around nearly as often as you’d like.

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 odds against making a royal flush are 649,739-to-1. By comparison, the odds of making a straight flush, poker’s second strongest hand, are 0.00139%, with the odds against at 72,192.3-to-1. Calculating the odds of royal flush for Texas Hold’em requires different mathematics, as Texas Hold’em hands are made by choosing the best five.

This is because the odds of being dealt a royal flush or other top-ranking hands are quite small. If you want to know more about exact numbers and the math behind rare hands in Texas Hold’em, keep on reading. Committing these facts to memory will probably make those rare occasions when you actually do manage to make one of these hands even more exciting and profitable.

Royal Flush (Straight Flush)

A royal flush is essentially the best possible hand that you can get in poker, but it’s still just a straight flush. So the odds of being dealt a royal are exactly the same as being dealt any other straight flush. You just need to have two hole cards that can make a royal, i.e., anything between a 10 and an Ace.

In this article, we’ll focus on the instances where you’ve got to use both of your cards to make your straight flush since doing it with just one card is neither as rare nor as lucrative, as you’ll have a four-straight, four-flush board that most players will tread very lightly on.

For two cards to be able to make a straight flush on the flop, they need to be suited and connected. However, not all of the starting hands meeting these criteria have the same odds, as some of them have more options than the others. Below is the table showing chances of flopping a straight flush with different types of suited connectors.

Hand	Number of possible combos	Percentage
A2s	Just one (2-3-4)	0.005%
Suited connectors (i.e. 78s)	Four (4-5-6, 5-6-9, 6-9-10, 9-10-J)	0.02%
One-gappers (i.e. 68s)	Three (4-5-7, 5-7-9, 7-9-10)	0.015%
Two-gappers (i.e. 69s)	Two (5-7-8, 7-8-10)	0.001%

If you’ve been wondering why that straight flush never comes in when you decide to peel the flop with your suited connectors, the table above probably offers a bit of a new perspective. The odds of this happening, even with the best possible candidate, are very slim.

Making a Straight Flush on Turn or River

More often than flopping a straight flush, you’ll actually flop a straight flush draw. It will be either an open-ended draw (with two possible cards to complete it) or a gutshot draw where only one card from the whole deck can improve you to straight flush.

Once you do flop that draw, the odds of making the hand actually improve quite significantly, even though they’re still low.

Draw type	Chance to improve on the turn	Chance to improve by the river
Gutshot	2.1%	4.3%
Open-ender	4.3%	8.4%

So if you have an open-ended straight flush draw on the flop, you have decent odds of hitting by the river. If your opponents will let you and if you account for the implied odds when you do hit this highly unlikely hand, these types of draws are sometimes worth chasing.

Quads (Four of a Kind)

Quads are another rare Hold’em hand although not nearly as rare as a straight flush. You’ve probably seen quite a few of these during your time playing poker and have had them at least once or twice yourself as well.

Once again, I’ll focus on odds of making quads when you are dealt a pocket pair as that’s the kind of quads you want to have and that you can extract some value from.

First of all, if you run the numbers, the overall odds of making quads with a random pocket pair if you were to just deal five cards (flop, turn, and river) are 0.816%. This is not a very useful stat, though, as you’ll rarely get to see all five cards unless you improve on the flop (i.e., flop a set).

Below are some more useful numbers.

Probability of royal flush in texas holdem

Odds of flopping quads	Odds of turning quads (after flopping a set)	Odds of getting quads by the river (after flopping a set)
0.245%	2.1%	4.3%

If you compare these numbers with the aforementioned ones for the straight flush, you can see why you’ve likely seen quads much more often than straight flushes.

And, if you were wondering what the odds are of flopping quads with just one card (i.e., when not holding a pocket pair), that would be 0.001%.

In Conclusion

Rare poker hands are at the top of hand strength charts for a reason. They’re very hard to get and when you do get them, you’ll almost always end up winning a pot. Even when you do lose with one of these hands, you might end up a winner as many cardrooms and casinos offer nice bad beat jackpots to tackle these precise scenarios.

Having seen these numbers, you’ll probably have a somewhat better idea of how (un)likely you are to be dealt a straight flush or quads during any individual session. Admittedly, this information won’t help you get any better in flopping royals. However, knowing the exact odds could save you a lot of money in the long run as you’ll know when to hold’em and when to fold’em.

What are pot odds? What about flushes and straights?

Basically, just forget about the name if you haven't heard about it before, there's no need to let it throw you off. Just think of 'pot odds' as the method for finding out whether chasing after a draw (like a flush or straight) is going to be profitable. If you're on your toes, you might have already been able to guess that it is generally better to chase after a draw when the bet is small rather than large, but we'll get to that in a minute...

Pot odds will tell you whether or not to call certain sized bets to try and complete your flush or straight draw.

Why use pot odds?

Because it makes you money, of course.

If you always know whether the best option is to fold or call when you're stuck with a hand like a flush draw, you are going to be saving (and winning) yourself money in the long run. On top of that, pot odds are pretty simple to work out when you get the hang of it, so it will only take a split second to work out if you should call or fold the next time you're in a sticky drawing situation. How nice is that?

How to work out whether or not to call with a flush or straight draw.

Now, this is the meat of the article. But trust me on this one, the 'working-out' part is not as difficult as you might think, so give me a chance to explain it to you before you decide to knock it on the head. So here we go...

Essentially, there are two quick and easy parts to working out pot odds. The first is to work out how likely it is that you will make your flush or straight (or whatever the hell you are chasing after), and the second is to compare the size of the bet that you are facing with the size of the pot. Then we use a little bit of mathematical magic to figure out if we should make the call.

1] Find out how likely it is to complete your draw (e.g. completing a flush draw).

All we have to do for this part is work out how many cards we have not seen, and then figure out how many of these unknown cards could make our draw and how many could not.

We can then put these numbers together to get a pretty useful ratio. So, for example, if we have a diamond flush draw on the flop we can work out...

The maths.

There are 47 cards that we do not know about (52 minus the 2 cards we have and minus the 3 cards on the flop).

9 of these unknown cards could complete our flush (13 diamonds in total minus 2 diamonds in our hand and the 2 diamonds on the flop).
The other 38 cards will not complete our flush (47 unknown cards, minus the helpful 9 cards results in 38 useless ones).
This gives us a ratio of 38:9, or scaled down... roughly 4:1.

So, at the end of all that nonsense we came out with a ratio of 4:1. This result is a pretty cool ratio, as it tells us that for every 4 times we get a useless card and miss our draw, 1 time will we get a useful card (a diamond) and complete our flush. Now all we need to do is put this figure to good use by comparing it to a similar ratio regarding the size of the bet that we are facing.

After you get your head around working out how many cards will help you and how many won't, the only tricky part is shortening a ratio like 38:9 down to something more manageable like 4:1. However, after you get used to pot odds you will just remember that things like flush draws are around 4:1 odds. To be honest, you won't even need to do this step the majority of the time, because there are very few ratios that you need to remember, so you can pick them off the top of your head and move on to step 2.

Texas Holdem Royal Flush Probability Rules

2] Compare the size of the bet to the size of the pot.

The title pretty much says it all here. Use your skills from the last step to work out a ratio for the size of the bet in comparison to the size of the pot. Just put the total pot size (our opponent's bet + the original pot) first in the ratio, and the bet size second. Here are a few quick examples for you...

Texas holdem royal flush probability calculator

$20 bet into a $100 pot = 120:20 = 6:1
$0.25 bet creating a total pot size of $1 = 1:0.25 = 4:1
$40 bet creating a total pot size of $100 = 100:40 = 2.5:1

That should be enough to give you an idea of how to do the second step. In the interest of this example, I am going to say that our opponent (with a $200 stack) has bet $20 in to a $80 pot, giving us odds of 5:1 ($100:$20). This is going to come in very handy in the next step.

This odds calculation step is very simple, and the only tricky part is getting the big ratios down into more manageable ones. However, this gets a lot easier after a bit of practice, so there's no need to give up just yet if you're not fluent when it comes to working with ratios after the first 5 seconds. Give yourself a chance!

To speed up your pot odds calculations during play, try using the handy (and free) SPOC program.

3] Compare these two ratios.

Now then, we know how likely it is that we are going to complete our draw, and we have worked out our odds from the pot (pot odds, get it? It's just like magic I know.). All we have to do now is put these two ratios side to side and compare them...

5:1 pot odds
4:1 odds of completing our draw on the next card

The pot odds in this case are bigger than the odds of completing our draw, which means that we will be making more money in the long run for every time we hit according to these odds. Therefore we should CALL because we will win enough to make up for the times that we miss and lose our money.

If that doesn't make total sense, then just stick to these hard and fast rules if it makes things easier:

If your pot odds are bigger than your chances of hitting - CALL
If your pot odds are smaller than your chances of hitting - FOLD

So just think of bigger being better when it comes to pot odds. Furthermore, if you can remember back to the start of the article when we had the idea that calling smaller bets is better, you will be able to work out that small bets give you bigger pot odds - makes sense right? It really comes together quite beautifully after you get your head around it.

What if there are two cards to come?

In this article I have shown you how to work out pot odds for the next card only. However, when you are on the flop there are actually 2 cards to come, so shouldn't you work out the odds for improving to make the best hand over the next 2 cards instead of 1?

No, actually.

Even if there are 2 cards to come (i.e. you're on the flop), you should still only work out the odds of improving your hand for the next card only.

The reason for this is that if you work using odds for improving over two cards, you need to assume that you won't be paying any more money on the turn to see the river. Seeing as you cannot be sure of this (it's quite unlikely in most cases), you should work out your pot odds for the turn and river individually. This will save you from paying more money than you should to complete your draw.

I discuss this important principle in a little more detail on my page about the rule of 2 and 4 for pot odds. It's also one of the mistakes poker players make when using odds.

Note: The only time you use odds for 2 cards to come combined is when your opponent in all-in on the flop. In almost every other case, you take it one card at a time.

Ultimate Texas Holdem Royal Flush Odds

Playing flush and straight draws overview.

I really tried hard to keep this article as short as possible, but then again I didn't want to make it vague and hazy so that you had no idea about what was going on. I'm hoping that after your first read-through that you will have a rough idea about how to work out when you should call or fold when on a flush or straight draw, but I am sure that it will take you another look over or two before it really starts to sink in. So I advise that you read over it again at least once.

The best way to get to grips with pot odds is to actually start working them out for yourself and trying them out in an actual game. It is all well and good reading about it and thinking that you know how to use them, but the true knowledge of pot odds comes from getting your hands dirty and putting your mind to work at the poker tables.

It honestly isn't that tough to use pot odds in your game, as it will take less than a session or two before you can use them comfortably during play. So trust me on this one, it is going to be well worth your while to spend a little time learning how to use pot odds, in return for always knowing whether to call or fold when you are on a draw. It will take a load off your mind and put more money in your pocket.

To help you out when it comes to your calculations, take a look at the article on simple pot odds. It should make it all a lot less daunting.

Go back to the sublime Texas Hold'em guide.

Probability Of A Royal Flush In Texas Holdem

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