Chances Of Hitting A Royal Flush In Texas Holdem

What are the odds of hitting a royal flush on a video poker machine? The odds of hitting a royal flush directly are only 1 in 649,739. But since you can draw one time your odds increase. If you play perfectly your odds of hitting a royal flush are roughly 1 in 40,000. The probability of not having a royal would be 1-0.000092=0.999908. An estimate of the probability there are no royals in 9 hands is 0.999908^9 = 0.999169. So the probability of at least one royal in 9 hands is 1-0.999169=0.000831.

Mathematics: Flushes & Straights : Simple Pot Odds : Implied Odds : Reverse Implied Odds

Watch SplitSuit's video on Flushes and Flush Draws for 8 hand histories involving strategy on playing flushes in Texas Hold'em.

You are on the flop with a pretty decent flush draw. You have two hearts in your hand and there are another two on the flop.

Unfortunately, some cool cat has made a bet, putting you in a tricky situation where you have to decide whether or not it is in your best interest to call to try and make the flush, or fold and save your money.

This is a prime example of where you are going to take advantage of 'pot odds' to work out whether or not it is worth making the call.

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.

Chances Of Hitting A Royal Flush In Texas Holdemlush In Texas Hold Em

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.

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...

• $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:

Chances Of Hitting A Royal Flush In Texas Holdemexas Hold Em

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.

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.

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Comments

For the benefit of others, in Omaha the player gets four hole cards, and there are five community cards. The player must use exactly two of his hole cards and three community cards.
I show the probability that any one player gets a royal is 42,807,600 /463,563,500,400 = 0.000092. The probability of not having a royal would be 1-0.000092=0.999908. An estimate of the probability there are no royals in 9 hands is 0.999908^9 = 0.999169. So the probability of at least one royal in 9 hands is 1-0.999169=0.000831.
Before somebody mentions it, this assumes that each hand is dealt from a separate deck. The probabilities are correlated when they all come from the same deck, but I think that effect is very minor.

For the benefit of others, in Omaha the player gets four hole cards, and there are five community cards. The player must use exactly two of his hole cards and three community cards.
I show the probability that any one player gets a royal is 42,807,600 /463,563,500,400 = 0.000092. The probability of not having a royal would be 1-0.000092=0.999908. An estimate of the probability there are no royals in 9 hands is 0.999908^9 = 0.999169. So the probability of at least one royal in 9 hands is 1-0.999169=0.000831.
Before somebody mentions it, this assumes that each hand is dealt from a separate deck. The probabilities are correlated when they all come from the same deck, but I think that effect is very minor.

Of course, this assumes that every player will play every hand all the way to the river, which is never the case. Using a fairly generous assumption that five out of nine players see the flop, and that two such players, on the average, play all the way to the river (many pots are resolved before then), you would have to multiply the figure above by ( (.55)(.4) ) to get a real-world estimation.

Of course, this assumes that every player will play every hand all the way to the river, which is never the case. Using a fairly generous assumption that five out of nine players see the flop, and that two such players, on the average, play all the way to the river (many pots are resolved before then), you would have to multiply the figure above by ( (.55)(.4) ) to get a real-world estimation.

I'm not sure which 'figure above' you are referring to. The way I would make that adjustment is to estimate the number out of 9 players who will stay in the whole way. Let's say that is 4 players. Then instead of the ^9 in my math above, do a ^4. I've never played a hand of Omaha in my life, so that 4 is just as example. To take it to another level, one should assume that with a possible royal, it is more likely the player will stay in than with random cards.

I'm not sure which 'figure above' you are referring to. The way I would make that adjustment is to estimate the number out of 9 players who will stay in the whole way. Let's say that is 4 players. Then instead of the ^9 in my math above, do a ^4. I've never played a hand of Omaha in my life, so that 4 is just as example. To take it to another level, one should assume that with a possible royal, it is more likely the player will stay in than with random cards.

The final figure: 0.000831, or whatever it was; I can't go back to that screen without losing this post.
If you figure that each hand, nine 'possible royals' are dealt, then you lose four of them as players fold preflop. (I know that most dealt hands don't contain possible royals at all, since they would have to have two suited cards ten or higher, but let's ignore that for the moment.) The play of the hand eliminates three out of those five before the river (and not every possible royal would play that far; if the flop only contains one helping card, then the player might still fold even though he would have gotten a runner-runner royal if he had stayed).
That is why I used the .55*.4 calculation. 5/9 of the players will play their hands, and 2/5 of those will play their hands all the way to the river.
It would be interesting to know how many Omaha hands contain 'possible royals'. Such hands would have to contain exactly two cards 10-A of a given suit, and could not contain a third such card . A hand could contain two such pairs of cards in different suits, of course. You could then assume, just for grins, that any such hand would play to the river, given that the flop would necessarily contain at least one helping card. That might give the best estimation of the 'chances for a royal'.
The reason I think this might be different from your previous result is that the majority of Omaha hands aren't even in the running for a royal from the start, so perhaps the best way to look at it might be from (percentage of Omaha hands that can make a royal)*(chance of completing the royal with five additional cards).

The final figure: 0.000831, or whatever it was; I can't go back to that screen without losing this post.
If you figure that each hand, nine 'possible royals' are dealt, then you lose four of them as players fold preflop. (I know that most dealt hands don't contain possible royals at all, since they would have to have two suited cards ten or higher, but let's ignore that for the moment.) The play of the hand eliminates three out of those five before the river (and not every possible royal would play that far; if the flop only contains one helping card, then the player might still fold even though he would have gotten a runner-runner royal if he had stayed).
That is why I used the .55*.4 calculation. 5/9 of the players will play their hands, and 2/5 of those will play their hands all the way to the river.
It would be interesting to know how many Omaha hands contain 'possible royals'. Such hands would have to contain exactly two cards 10-A of a given suit, and could not contain a third such card . A hand could contain two such pairs of cards in different suits, of course. You could then assume, just for grins, that any such hand would play to the river, given that the flop would necessarily contain at least one helping card. That might give the best estimation of the 'chances for a royal'.
The reason I think this might be different from your previous result is that the majority of Omaha hands aren't even in the running for a royal from the start, so perhaps the best way to look at it might be from (percentage of Omaha hands that can make a royal)*(chance of completing the royal with five additional cards).

so basically, once every 1200 dealt hands at a full Omaha table.

That is why I used the .55*.4 calculation. 5/9 of the players will play their hands, and 2/5 of those will play their hands all the way to the river.

That is not the correct way to do it. Suppose you ask what is the probability that if 10 people toss a coin 5 times, that at least one of them will get 5 tails? That answer is 1-(1-(1/2)^5)^10 = 1-(31/32)^10 = 1-.72798 = .27202.
No suppose half of them drop out before finishing. I think by your logic you would say the probability drops to .5*.27202 = .13601.
How it in fact changes to 1-(1-(1/2)^5)^5 = 1-(31/32)^5 = 1-.853215 = .146785.
Close, but no cigar.
Agreed that not all 10 people can get a royal. I admitted it was an estimate.

That is not the correct way to do it. Suppose you ask what is the probability that if 10 people toss a coin 5 times, that at least one of them will get 5 tails? That answer is 1-(1-(1/2)^5)^10 = 1-(31/32)^10 = 1-.72798 = .27202.
No suppose half of them drop out before finishing. I think by your logic you would say the probability drops to .5*.27202 = .13601.
How it in fact changes to 1-(1-(1/2)^5)^5 = 1-(31/32)^5 = 1-.853215 = .146785.
Close, but no cigar.
Agreed that not all 10 people can get a royal. I admitted it was an estimate.

If you only need one occurence of an event to fulfill a desired condition, aren't the probabilities additive? So if a nine-handed game has X probability of one player winding up with a royal, doesn't that mean that any single player has X/9 probability of hitting a royal--and if five people at least see the flop, then the chances of the table hitting a royal are 5*(X/9)?

If you only need one occurence of an event to fulfill a desired condition, aren't the probabilities additive? So if a nine-handed game has X probability of one player winding up with a royal, doesn't that mean that any single player has X/9 probability of hitting a royal--and if five people at least see the flop, then the chances of the table hitting a royal are 5*(X/9)?

Probabilities are only additive if the deseried condition must happen withing so many trials. For example drawing the ace of spades from a single deck of cards. The probability of getting it in 10 cards is 10x that of one card.

Probabilities are only additive if the deseried condition must happen withing so many trials. For example drawing the ace of spades from a single deck of cards. The probability of getting it in 10 cards is 10x that of one card.

The main point of the question was to determine when the royal flush jackpot at UB becomes a good bet.

Odds Of Hitting A Straight Flush In Texas Holdem

The jackpot tables rake a portion of any pot over 10 BB, up to 1 BB.
65% of the jackpot goes to the following distribution:
50% to the RF holder
25% to the rest of the players at the table
25% to the rest of the players at the same stakes.
25% of the remainder feeds the next jackpot, 10% to UB.

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