In our poker math and probability lesson it was stated that when it comes to poker; “the math is essential“. Although you don’t need to be a math genius to play poker, a solid understanding of probability will serve you well and knowing the odds is what it’s all about in poker. It has also been said that in poker, there are good bets and bad bets. The game just determines who can tell the difference. That statement relates to the importance of knowing and understanding the math of the game.

In this lesson, we’re going to focus on drawing odds in poker and how to calculate your chances of hitting a winning hand. We’ll start with some basic math before showing you how to correctly calculate your odds. Don’t worry about any complex math – we will show you how to crunch the numbers, but we’ll also provide some simple and easy shortcuts that you can commit to memory.

Dec 25, 2019  Poker Tips & Strategy The Odds of Different Rare Card Hands Author: Ivan Potocki December 25, 2019. 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. Odds of flopping a straight flush draw with two suited cards between Ten and Ace = 141/19,600 = 0.0072 or roughly 0.72%. That’s still less than 1% chance of flopping a Royal Flush draw even after starting out with a suited connector, but it is 141 times more likely than flopping the Royal Flush itself.

Basic Math – Odds and Percentages

Odds can be expressed both “for” and “against”. Let’s use a poker example to illustrate. The odds against hitting a flush when you hold four suited cards with one card to come is expressed as approximately 4-to-1. This is a ratio, not a fraction. It doesn’t mean “a quarter”. To figure the odds for this event simply add 4 and 1 together, which makes 5. So in this example you would expect to hit your flush 1 out of every 5 times. In percentage terms this would be expressed as 20% (100 / 5).

Here are some examples:

• 2-to-1 against = 1 out of every 3 times = 33.3%
• 3-to-1 against = 1 out of every 4 times = 25%
• 4-to-1 against = 1 out of every 5 times= 20%
• 5-to-1 against = 1 out of every 6 times = 16.6%

Converting odds into a percentage:

• 3-to-1 odds: 3 + 1 = 4. Then 100 / 4 = 25%
• 4-to-1 odds: 4 + 1 = 5. Then 100 / 5 = 20%

Converting a percentage into odds:

• 25%: 100 / 25 = 4. Then 4 – 1 = 3, giving 3-to-1 odds.
• 20%: 100 / 20 = 5. Then 5 – 1 = 4, giving 4-to-1 odds.

Another method of converting percentage into odds is to divide the percentage chance when you don’t hit by the percentage when you do hit. For example, with a 20% chance of hitting (such as in a flush draw) we would do the following; 80% / 20% = 4, thus 4-to-1. Here are some other examples:

• 25% chance = 75 / 25 = 3 (thus, 3-to-1 odds).
• 30% chance = 70 / 30 = 2.33 (thus, 2.33-to-1 odds).

Some people are more comfortable working with percentages rather than odds, and vice versa. What’s most important is that you fully understand how odds work, because now we’re going to apply this knowledge of odds to the game of poker.

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Counting Your Outs

Before you can begin to calculate your poker odds you need to know your “outs”. An out is a card which will make your hand. For example, if you are on a flush draw with four hearts in your hand, then there will be nine hearts (outs) remaining in the deck to give you a flush. Remember there are thirteen cards in a suit, so this is easily worked out; 13 – 4 = 9.

Another example would be if you hold a hand like and hit two pair on the flop of . You might already have the best hand, but there’s room for improvement and you have four ways of making a full house. Any of the following cards will help improve your hand to a full house; .

The following table provides a short list of some common outs for post-flop play. I recommend you commit these outs to memory:

Table #1 – Outs to Improve Your Hand

The next table provides a list of even more types of draws and give examples, including the specific outs needed to make your hand. Take a moment to study these examples:

Table #2 – Examples of Drawing Hands (click to enlarge)

Counting outs is a fairly straightforward process. You simply count the number of unknown cards that will improve your hand, right? Wait… there are one or two things you need to consider:

Don’t Count Outs Twice

There are 15 outs when you have both a straight and flush draw. You might be wondering why it’s 15 outs and not 17 outs, since there are 8 outs to make a straight and 9 outs for a flush (and 8 + 9 = 17). The reason is simple… in our example from table #2 the and the will make a flush and also complete a straight. These outs cannot be counted twice, so our total outs for this type of draw is 15 and not 17.

Anti-Outs and Blockers

There are outs that will improve your hand but won’t help you win. For example, suppose you hold on a flop of . You’re drawing to a straight and any two or any seven will help you make it. However, the flop also contains two hearts, so if you hit the or the you will have a straight, but could be losing to a flush. So from 8 possible outs you really only have 6 good outs.

It’s generally better to err on the side of caution when assessing your possible outs. Don’t fall into the trap of assuming that all your outs will help you. Some won’t, and they should be discounted from the equation. There are good outs, no-so good outs, and anti-outs. Keep this in mind.

Calculating Your Poker Odds

Once you know how many outs you’ve got (remember to only include “good outs”), it’s time to calculate your odds. There are many ways to figure the actual odds of hitting these outs, and we’ll explain three methods. This first one does not require math, just use the handy chart below:

Table #3 – Poker Odds Chart

As you can see in the above table, if you’re holding a flush draw after the flop (9 outs) you have a 19.1% chance of hitting it on the turn or expressed in odds, you’re 4.22-to-1 against. The odds are slightly better from the turn to the river, and much better when you have both cards still to come. Indeed, with both the turn and river you have a 35% chance of making your flush, or 1.86-to-1.

We have created a printable version of the poker drawing odds chart which will load as a PDF document (in a new window). You’ll need to have Adobe Acrobat on your computer to be able to view the PDF, but this is installed on most computers by default. We recommend you print the chart and use it as a source of reference. It should come in very handy.

Doing the Math – Crunching Numbers

There are a couple of ways to do the math. One is complete and totally accurate and the other, a short cut which is close enough.

Let’s again use a flush draw as an example. The odds against hitting your flush from the flop to the river is 1.86-to-1. How do we get to this number? Let’s take a look…

With 9 hearts remaining there would be 36 combinations of getting 2 hearts and making your flush with 5 hearts. This is calculated as follows:

(9 x 8 / 2 x 1) = (72 / 2) ≈ 36.

This is the probability of 2 running hearts when you only need 1 but this has to be figured. Of the 47 unknown remaining cards, 38 of them can combine with any of the 9 remaining hearts:

9 x 38 ≈ 342.

Now we know there are 342 combinations of any non heart/heart combination. So we then add the two combinations that can make you your flush:

36 + 342 ≈ 380.

The total number of turn and river combos is 1081 which is calculated as follows:

(47 x 46 / 2 x 1) = (2162 / 2) ≈ 1081.

Now you take the 380 possible ways to make it and divide by the 1081 total possible outcomes:

380 / 1081 = 35.18518%

This number can be rounded to .352 or just .35 in decimal terms. You divide .35 into its reciprocal of .65:

0.65 / 0.35 = 1.8571428

And voila, this is how we reach 1.86. If that made you dizzy, here is the short hand method because you do not need to know it to 7 decimal points.

The Rule of Four and Two

A much easier way of calculating poker odds is the 4 and 2 method, which states you multiply your outs by 4 when you have both the turn and river to come – and with one card to go (i.e. turn to river) you would multiply your outs by 2 instead of 4.

Imagine a player goes all-in and by calling you’re guaranteed to see both the turn and river cards. If you have nine outs then it’s just a case of 9 x 4 = 36. It doesn’t match the exact odds given in the chart, but it’s accurate enough.

What about with just one card to come? Well, it’s even easier. Using our flush example, nine outs would equal 18% (9 x 2). For a straight draw, simply count the outs and multiply by two, so that’s 16% (8 x 2) – which is almost 17%. Again, it’s close enough and easy to do – you really don’t have to be a math genius.

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Conclusion

In this lesson we’ve covered a lot of ground. We haven’t mentioned the topic of pot odds yet – which is when we calculate whether or not it’s correct to call a bet based on the odds. This lesson was step one of the process, and in our pot odds lesson we’ll give some examples of how the knowledge of poker odds is applied to making crucial decisions at the poker table.

As for calculating your odds…. have faith in the tables, they are accurate and the math is correct. Memorize some of the common draws, such as knowing that a flush draw is 4-to-1 against or 20%. The reason this is easier is that it requires less work when calculating the pot odds, which we’ll get to in the next lesson.

By Tom 'TIME' Leonard

Tom has been writing about poker since 1994 and has played across the USA for over 40 years, playing every game in almost every card room in Atlantic City, California and Las Vegas.

The odds of flopping a straight flush with a premium suited connector such as T9s is 0.02% or 1 in 4,900

Definition of the Straight Flush –

Five cards of consecutive rank, all of the same suit.

Example – 5d6d7d8d9d

The Ten to Ace Straight Flush is the strongest hand in poker and is referred to as the “Royal Flush”.

Odds of Making a Straight Flush on the Flop

Flopping a Straight Flush seldom happens in poker. We specifically need to start out with two suited connected cards for this to be possible.

The odds of flopping a Straight Flush are so unlikely (0.02% or less) that the majority of poker equity calculators don’t even show the precise odds.

We’ll need to do some maths of our own.

Calculation of Straight Flush Odds

Let’s start with a very specific example -

We hold A2s. What are the odds of flopping the Ace to Five Straight Flush?

Why do we choose this example? It’s the easiest one because it provides only one way of making the Straight Flush. The flop has to come down precisely Three, Four, Five of the correct suit.

So, how likely is this?

In order to calculate, we’ll first need to know how many combinations of three cards are possible on the flop.

Basic Combinations and Permutations

Firstly, how many different combinations of three cards can be dealt on the flop? Assuming we care about the order of the three cards (and that our two hole cards are already known), the answer is 117,600 (50 * 49 * 48).

In statistics, this type of calculation is referred to as a permutation and accounts for the order of the flop cards.

Of course, in Hold’em, the order of the cards on the flop doesn’t matter (i.e. a 3,4,5 flop is the same as a 5,3,4 flop, for all intents and purposes). What we are interested in is the number of possible combinations of three cards.

A combination is similar to a permutation but doesn’t account for the order. Since there are 6 possible ways of arranging three cards, we can simply divide our number of permutations (117,600) by 6 to establish the number of possible three-card combinations on the flop.

117,600 / 6 = 19,600 possible combinations of three cards on the flop (given two cards are known)

In other words, there are 19,600 different possible sets of three cards that may fall on the flop given that our two hole cards are already known.

Guess what?

To make the exact Straight Flush in question, only one of these 19,600 combinations will do the job.

Armed with that information, we can now establish a range of different probabilities.

Odds of flopping the Straight Flush with A2s = 1/19,600 = 0.00005 or roughly 0.005%

That’s an insanely small likelihood!

Thankfully, the odds with different types of starting hands are usually a little better.

It all depends on the number of different combinations of three cards that provide a Straight Flush.

For example, think about T9s.

How many different ways are there to make a Straight Flush with 9Ts?

Ways of making a Straight Flush with T9s

Odds Of A Straight Flush

JQK
QJ8
J87
678

So that’s four different ways. We are hence four times as likely to make a Straight Flush with 9Ts as we are to make a Straight Flush with A2s.

Odds of flopping the Straight Flush with 9Ts = 4/19,600 = 0.0002 or roughly 0.02%

Ways of making a Straight Flush with T8s

QJ9
J79
679

Odds of flopping the Straight Flush with T8s = 3/19,600 = 0.00015 or roughly 0.015%

Ways of making a Straight Flush with T7s

Flush Vs Straight Odds

J89
689

Odds of flopping the Straight Flush with T7s = 2/19,600 = 0.0001 or roughly 0.01%

Only suited connectors (or gappers) can make Straight Flushes on the flop. All other holdings such as pocket-pairs and off-suit combos can never flop a Straight Flush.

We are, naturally, more likely to flop a Straight Flush draw as opposed to the Straight Flush itself. To see examples of calculating the odds of hitting a Straight Flush draw on the flop, check out the 888poker article on Royal Flush odds in poker.

Odds of Making a Straight Flush on the Later Streets

There will be two primary types of Straight Flush draw we’ll flop. The gutshot Straight Flush draw and the open-ended Straight Flush draw.

Gutshot Straight Flush draws have 1 out in the deck, while open ended Straight Flush draws have 2 outs in the deck.

Odds of Hitting on the Turn or River

Odds of catching the gutshot Straight Flush on the turn 1/47 = 0.0213 or roughly 2.1%

Odds of catching the open-ended Straight Flush on the turn 2/47 = 0.426 or roughly 4.3%

Odds of catching the gutshot Straight Flush on the river 1/46 = 0.0217 or roughly 2.2%

Odds of catching the open-ended Straight Flush on the river 2/46 = 0.0435 or roughly 4.4%

Odds of Hitting by the River

To calculate the probability of hitting by the river, we’ll employ the trick of calculating the possibility of not hitting and then subtracting from 100%.

Odds of not catching the gutshot Straight Flush on the turn 46/47

Odds of not catching the open-ended Straight Flush on the turn 45/47

Odds of not catching the gutshot Straight Flush on the river 45/46

Odds of not catching the open-ended Straight Flush on the river 44/46

Odds of not catching the gutshot Straight Flush on the turn or river = 46/47 * 45/46 = 0.9574 or roughly 95.7%

Odds of not catching the open-ended Straight Flush on the turn or river = 45/47 * 44/46 = 0.9158 or roughly 91.6%

Odds of hitting the gutshot Straight Flush by the river = (100- 95.7%) roughly 4.3%

Odds of hitting the open-ended Straight Flush by the river = (100 – 91.6%) roughly 8.4%

Implied Odds Analysis of a Straight Flush

A Straight Flush always carries excellent implied odds when hitting. This is because our opponent is usually forced into stacking off with very strong worse hands such as worse flushes and full houses.

Straight Flushes made with two of our hole cards always carry better implied odds than Straight Flushes made with one of our hole cards.

When using just one of our hole cards, it means there will be four cards to the Straight Flush already on the board. This decreases the chance that our opponent will pay us off with worse holdings.

Although Straight Flushes should hardly ever be folded, their implied odds are the best when no higher Straight Flush is possible on the board.

Basic Strategy Advice

Odds Of Flopping Two Pair

It’s basically the nuts. Play aggressively and make big bets! Even if a higher Straight Flush is possible, it’s usually just a cooler if we are beat. We’d have to be really deep to find an exception.

Odds Of Flopping A Set

Odds of Making Straight Flush
Method (Straight Flush)	Probability (%)
Flopping the Straight Flush with A2s	0.01
Flopping the Straight Flush with T9s	0.02
Flopping the Straight Flush with T8s	0.02
Flopping the Straight Flush with T7s	0.02
Catching the Straight Flush Gutshot from flop to turn	2.13
Catching the Straight Flush open ender from flop to turn	4.26
Catching the Straight Flush Gutshot from turn to river	2.17
Catching the Straight Flush open ender from turn to river	4.35
Catching the Straight Flush Gutshot from flop to river	4.30
Catching the Straight Flush open ender from flop to river	8.42