Brian Alspach

• Poker Hand Probabilities. Mark Brader has provided the following tables of probabilities of the various five-card poker hands when five cards are dealt from a single 52-card deck, and also when using multiple decks. The traditional hand types are described on the poker hand ranking page. These include one hand that belongs to two types at once.
• Know Your Poker Odds. We’ll get you started by showing you 20 examples of the basic Texas Hold’em odds you need to know. To really make a mark on the felt, we’re also going to show you a nifty little trick for calculating poker odds right at the table.
• Probability of Poker Hands Drew Armstrong armstron@math.umn.edu November 1, 2006 In a standard deck of cards, there are 4 possible suits (clubs, diamonds, hearts, spades), and 13 possible values (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace). Let A,J,Q,K represent Ace, Jack, Queen and King, respectively.

18 January 2000

For more information on poker odds and winning at poker, try the following links: To see all of our articles on poker rules and advice, go to our main article on How To Play Poker. For an introduction to the game, skim over these Poker Basics. So you think you've got the best hand. Maximize your winnings with these Poker Betting Tips.

Abstract:

One of the most popular poker games is 7-card stud. The way hands areranked is to choose the highest ranked 5-card hand contained amongst the7 cards. People frequently encounter difficulty in counting 7-card handsbecause a given set of 7 cards may contain several different types of5-card hands. This means duplicate counting can be troublesome as canomission of certain hands. The types of 5-card poker hands in decreasingrank are

• straight flush
• 4-of-a-kind
• full house
• flush
• straight
• 3-of-a-kind
• two pairs
• a pair
• high card

The total number of 7-card poker hands is .

We shall count straight flushes using the largest card in the straightflush. This enables us to pick up 6- and 7-card straight flushes. Whenthe largest card in the straight flush is an ace, then the 2 other cardsmay be any 2 of the 47 remaining cards. This gives us straight flushes in which the largest card is an ace.

If the largest card is any of the remaining 36 possible largest cards ina straight flush, then we may choose any 2 cards other than theimmediate successor card of the particular suit. This gives usstraight flushes of the second type, and41,584 straight flushes altogether.

In forming a 4-of-a-kind hand, there are 13 choices for the rank ofthe quads, 1 choice for the 4 cards of the given rank, and choices for the remaining 3 cards. This implies there are 4-of-a-kind hands.

There are 3 ways to get a full house and we count them separately. Oneway of obtaining a full house is for the 6-card hand to contain 2 setsof triples and a singleton. There are ways tochoose the 2 ranks, 4 ways to choose each of the triples, and 44 ways tochoose the singleton. This gives us fullhouses of this type. A second way of getting a full houseis for the 7-card hand to contain a triple and 2 pairs. There are 13ways to choose the rank of the triple, ways tochoose the ranks of the pairs, 4 ways to choose the triple of the givenrank, and 6 ways to choose the pairs of each of the given ranks. Thisproduces full house of the secondkind. The third way to get a full house is for the 7-card hand tocontain a triple, a pair and 2 singletons of distinct ranks. There are13 choices for the rank of the triple, 12 choices for the rank of thepair, choices for the ranks of the singletons,4 choices for the triple, 6 choices for the pair, and 4 choices for eachof the singletons. We obtain full houses of the last kind. Adding the 3 numbers gives us3,473,184 full houses.

To count the number of flushes, we first obtain some useful informationon sets of ranks. The number of ways of choosing 7 distinct ranks from13 is .We want to remove the sets of rankswhich include 5 consecutive ranks (that is, we are removing straightpossibilities). There are 8 rank sets of the form .Another form to eliminate is ,where y is neither x-1 nor x+6. If x is ace or 9, thereare 6 choices for y. If x is any of the other 7 possibilities, thereare 5 possibilities for y. This produces sets with 6 consecutive ranks. Finally, the remaining form to eliminateis ,where neither y nor z is allowed totake on the values x-1 or x+5. If x is either ace or 10, theny,z are being chosen from a 7-subset. If x is any of the other 8possible values, then y,z are being chosen from a 6-set. Hence, thenumber of rank sets being excluded in this case is .In total, we remove 217 sets of ranks ending upwith 1,499 sets of 7 ranks which do not include 5 consecutive ranks.Thus, there are flushes having all 7 cards in thesame suit.

Now suppose we have 6 cards in the same suit. Again there are 1,716sets of 6 ranks for these cards in the same suit. We must excludesets of ranks of the form of which thereare 9. We also must exclude sets of ranks of the form ,where y is neither x-1 nor x+5. So if x is aceor 10, y can be any of 7 values; whereas, if x is any of the other8 possible values, y can be any of 6 values. This excludes 14 + 48= 62 more sets. Altogether 71 sets have been excluded leaving 1,645sets of ranks for the 6 suited cards not producing a straight flush.The remaining card may be any of the 39 cards from the other 3 suits.This gives us flushes with 6 suitedcards.

Finally, suppose we have 5 cards in the same suit. The remaining 2cards cannot possibly give us a hand better than a flush so all we needdo here is count flushes with 5 cards in the same suit. There arechoices for 5 ranks in the same suit. We mustremove the 10 sets of ranks producing straight flushes leaving us with1,277 sets of ranks. The remaining 2 cards can be any 2 cards from theother 3 suits so that there are choices for them.Then there are flushes of this lasttype. Adding the numbers of flushes of the 3 types produces 4,047,644flushes.

We saw above that there are 217 sets of 7 distinct ranks which include5 consecutive ranks. For any such set of ranks, each card may be anyof 4 cards except we must remove those which correspond to flushes.There are 4 ways to choose all of them in the same suit. There areways to choose 6 of them in the same suit. For 5of them in the same suit, there are ways to choosewhich 5 will be in the same suit, 4 ways to choose the suit of the 5cards, and 3 independent choices for the suits of each of the 2 remainingcards. This gives choices with 5 in the samesuit. We remove the 844 flushes from the 4⁷ = 16,384 choices of cardsfor the given rank set leaving 15,540 choices which produce straights.We then obtain straights when the 7-cardhand has 7 distinct ranks.

We now move to hands with 6 distinct ranks. One possible form is,where x can be any of 9 ranks. The otherpossible form is ,where y is neither x-1nor x+5. When x is ace or 10, then there are 7 choices for y.When x is between 2 and 9, inclusive, there are 6 choices for y.This implies there are sets of 6 distinctranks corresponding to straights. Note this means there must be a pairin such a hand. We have to ensure we do not count any flushes.

As we just saw, there are 71 choices for the set of 6 ranks. Thereare 6 choices for which rank will have a pair and there are 6 choicesfor a pair of that rank. Each of the remaining 5 cards can be chosenin any of 4 ways. Now we remove flushes. If all 5 cards were chosenin the same suit, we would have a flush so we remove the 4 ways ofchoosing all 5 in the same suit. In addition, we cannot choose 4 ofthem in either suit of the pair. There are 5 ways to choose 4 cardsto be in the same suit, 2 choices for that suit and 3 choices for thesuit of the remaining card. So there are choices which give a flush. This means there are 4⁵ - 34 = 990choices not producing a flush. Hence, there are straights of this form.

We also can have a set of 5 distinct ranks producing a straight whichmeans the corresponding 7-card hand must contain either 2 pairs or3-of-a-kind as well. The set of ranks must have the formand there are 10 such sets. First we supposethe hand also contains 3-of-a-kind. There are 5 choices for the rankof the trips, and 4 choices for trips of that rank. The cards of theremaining 4 ranks each can be chosen in any of 4 ways. This gives4⁴ = 256 choices for the 4 cards. We must remove the 3 choices for whichall 4 cards are in the same suit as one of the cards in the 3-of-a-kind.So we have straights which alsocontain 3-of-a-kind.

Next we suppose the hand also contains 2 pairs. There are choices for the 2 ranks which will be paired. There are 6choices for each of the pairs giving us 36 ways to choose the 2 pairs.We have to break down these 36 ways of getting 2 pairs because differentsuit patterns for the pairs allow different possibilities for flushesupon choosing the remaining 3 cards. Now 6 of the ways of getting the2 pairs have the same suits represented for the 2 pairs, 24 of themhave exactly 1 suit in common between the 2 pairs, and 6 of them haveno suit in common between the 2 pairs.

There are 4³ = 64 choices for the suits of the remaining 3 cards.In the case of the 6 ways of getting 2 pairs with the same suits, 2of the 64 choices must be eliminated as they would produce a flush(straight flush actually). In the case of the 24 ways of getting 2pairs with exactly 1 suit in common, only 1 of the 64 choices need beeliminated. When the 2 pairs have no suit in common, all 64 choicesare allowed since a flush is impossible. Altogether we obtain

straights which alsocontain 2 pairs. Adding all the numbers together gives us 6,180,020straights.

A hand which is a 3-of-a-kind hand must consist of 5 distinct ranks.There are sets of 5 distinct ranks fromwhich we must remove the 10 sets corresponding to straights. Thisleaves 1,277 sets of 5 ranks qualifying for a 3-of-a-kind hand. Thereare 5 choices for the rank of the triple and 4 choices for the tripleof the chosen rank. The remaining 4 cards can be assigned any of 4suits except not all 4 can be in the same suit as the suit of one ofcards of the triple. Thus, the 4 cards may be assigned suits in 4⁴-3=253 ways. Thus, we obtain 3-of-a-kind hands.

Next we consider two pairs hands. Such a hand may contain either 3pairs plus a singleton, or two pairs plus 3 remaining cards of distinctranks. We evaluate these 2 types of hands separately. If the hand has3 pairs, there are ways to choose the ranks ofthe pairs, 6 ways to choose each of the pairs, and 40 ways to choosethe singleton. This produces 7-card hands with 3 pairs.

The other kind of two pairs hand must consist of 5 distinct ranks andas we saw above, there are 1,277 sets of ranks qualifying for a twopairs hand. There are choices for the two ranksof the pairs and 6 choices for each of the pairs. The remaining cardsof the other 3 ranks may be assigned any of 4 suits, but we must removeassignments which result in flushes. This results in exactly thesame consideration for the overlap of the suits of the two pairs asin the final case for flushes above. We then obtain

hands of two pairs of the second type. Adding the two gives 31,433,4007-card hands with two pairs.

Now we count the number of hands with a pair. Such a hand must have6 distinct ranks. We saw above there are 1,645 sets of 6 ranks whichpreclude straights. There are 6 choices for the rank of the pair and6 choices for the pair of the given rank. The remaining 5 ranks canhave any of 4 suits assigned to them, but again we must remove thosewhich produce a flush. We cannot choose all 5 to be in the same suitfor this results in a flush. This can happen in 4 ways. Also, wecannot choose 4 of them to be in the same suit as the suit of eitherof the cards forming the pair. This can happen in ways. Hence, there are 4⁵-34 = 990 choices for the remaining 4 cards.This gives us hands with a pair.

We could determine the number of high card hands by removing the handswhich have already been counted in one of the previous categories.Instead, let us count them independently and see if the numbers sumto 133,784,560 which will serve as a check on our arithmetic.

A high card hand has 7 distinct ranks, but does not include straights.So we must eliminate sets of ranks which have 5 consecutive ranks.Above we determined there are 1,499 sets of 7 ranks not containing 5consecutive ranks, that is, there are no straights. Now the card ofeach rank may be assigned any of 4 suits giving 4⁷ = 16,384 assignmentsof suits to the ranks. We must eliminate those which resulkt in flushes.There are 4 ways to assign all 7 cards the same suit. There are 7choices for 6 cards to get the same suit, 4 choices of the suit to beassigned to the 6 cards, and 3 choices for the suit of the other card.This gives assignments in which 6 cards end upwith the same suit. Finally, there are choices for5 cards to get the same suit, 4 choices for that suit, and 3 independentchoices for each of the remaining 2 cards. This gives assignments producing 5 cards in the same suit. Altogether wemust remove 4 + 84 + 756 = 844 assignments resulting in flushes. Thus,the number of high card hands is 1,499(16,384 - 844)=23,294,460.

If we sum the preceding numbers, we obtain 133,784,560 and we can beconfident the numbers are correct.

Here is a table summarizing the number of 7-card poker hands. Theprobability is the probability of having the hand dealt to you whendealt 7 cards.

hand	number	Probability
straight flush	41,584	.00031
4-of-a-kind	224,848	.0017
full house	3,473,184	.026
flush	4,047,644	.030
straight	6,180,020	.046
3-of-a-kind	6,461,620	.048
two pairs	31,433,400	.235
pair	58,627,800	.438
high card	23,294,460	.174

You will observe that you are less likely to be dealt a hand withno pair (or better) than to be dealt a hand with one pair. Thishas caused some people to query the ranking of these two hands.In fact, if you were ranking 7-card hands based on 7 cards, theorder of the last 2 would switch. However, you are basing the rankingon 5 cards so that if you were to rank a high card hand higher than a handwith a single pair, people would choose to ignore the pair in a7-card hand with a single pair and call it a high card hand. Thiswould have the effect of creating the following distortion. Thereare 81,922,260 7-card hands in the last two categories containing5 cards which are high card hands. Of these 81,922,260 hands,58,627,800 also contain 5-card hands which have a pair. Thus, thelatter hands are more special and should be ranked higher (as theyindeed are) but would not be under the scheme being discussed inthis paragraph.

Winning Texas holdem poker players have to have a solid understanding of odds and pot odds.

Many inexperienced players make the mistake of assuming odds and pot odds are the same thing. While the two things are related, they aren’t the same.

The first thing you’re going to learn is what odds and pot odds are, how to quickly calculate them, and how to use them to improve your results at the table. Then you’ll find an extensive list of examples so you can practice what you’ve learned.

It’s a good idea to bookmark this page so you can come back and go over the examples frequently. The more times you go over them the better your ability will be to make the correct decisions at the holdem table.

The best way to develop strong odds and pot odds calculating skills is to study this page and then use what you’ve learned in live Texas holdem play. You’ll make mistakes as you play, but every time you make one make a mental note or write it down.

Then study these situations after the game to see where you made a mistake and learn how to avoid making the same mistake in the future.

A mistake that many new Texas holdem players make is not learning about odds and pot odds because they’re afraid of the math or that it’s too hard. While both items do involve some math, it isn’t difficult once you understand it.

More importantly, the most common odds and pot odds situations happen frequently so you’ll quickly memorize the important situations and won’t have to calculate many hands while playing.

You’ll also learn a few quick tricks and tips that the pros know to help you make fast decisions in the middle of a hand.

Odds

Odds are a mathematical way of explaining how likely or unlikely something is to happen.

You can use odds in two different formats. The first one is a percentage and the second is a ratio. Some players find percentages easier to work with, but you need to learn how to consider Texas holdem odds as ratios. This is important because when you start using pot odds if your odds are already in ratios it saves a step. If you have the odds in percentages you have to either convert the pot to a percentage or convert the odds from a percentage to a ratio.

If this seems confusing, don’t panic. The rest of the page covers odds and pot odds using ratios so you’ll learn the best way possible.

Here’s an example of how you use odds in Texas holdem.

If you have pocket queens and have already seen the flop and it has no queens, what are the odds a queen will land on the turn?

To determine the correct odds you need to determine how many unseen cards remain in the deck. You’ve seen your two hole cards and the three board cards. So you’ve seen five out of 52 cards, leaving 47 unseen cards. You also know that there are two more queens.

This means that two of the remaining cards will give you another queen and 45 of them won’t.

This gives a ratio of 2 to 45 or 45 to 2, depending on how you want to list it. Usually the odds against number is listed first, so you’ll see 45 to 2 or 45:2.

In simpler terms, this means that on average, if you play this exact situation 47 time that you’ll get a queen twice and a card other than a queen 45 times.

If a queen doesn’t land on the turn, what are the odds of one landing on the river?

You’ve seen one more card so two cards are still queens and 44 aren’t.

Here’s another example.

If you have four to a flush after the flop, what are the odds the turn will make a flush for you?

In this situation you have 9 cards out of 47 unseen that will complete your flush. So 9 cards complete your flush and 38 don’t. This means 38 to 9 or 38:9.

If you don’t complete your flush on the turn the odds change to 37:9 on the river.

But what about the cards in your opponent’s hands and the ones that are already in the muck?

Don’t make the mistake of thinking that because your opponents have seen their cards that you don’t count them in your calculations. The only numbers that matter are the number of cards you’ve seen and the number of cards you haven’t seen.

Sometimes the card or some of the cards you need are in other player’s hands or in the muck, but it doesn’t matter. The cards you need are just as likely to be in the remaining cards to be dealt and in the long run they’ll be in any particular location an equal number of times based on the probabilities.

The way many players visualize this is by assigning each place in the deck of cards a number, 1 through 52. As the deck is ready to have the first card dealt the top card is number 1 and the second card is number 2 down to the bottom card being number 52.

As the cards are dealt, burned, folded, etc. more cards are used from the top of the deck. It doesn’t matter what happens to each card or if you see it or not.

In the long run each card has an equal chance to be number 1, number 36, number 52, or any other number. Over millions of hands each individual card will be in each of the 52 places the same number of times.

This means that if you need the ace of clubs to complete your hand it’s an unseen card and doesn’t matter where it’s located during the current stack. Over the long run it’ll be in each of the 52 positions an equal number of times, so all you can do is work with the seen and unseen cards.

You can also use odds to determine the likelihood of other things happening during a hand.

What are the odds that the first card you’re dealt is an ace?

You know the deck has 52 cards and four aces, so the odds of the first card you receive being an ace are 48:4. This can be reduced to 12:1. You reduce ratios by dividing the same number into both numbers. In this case you divide both 48 and 4 by 4.

Another way to look at this is 1 out of every 13 cards on average is an ace. This makes sense because each suit has 13 cards and one of them is an ace.

Ratios are important because you use them when determining pot odds. You’ll learn more about pot odds in the next section.

But if you want to know the odds of things like getting dealt pocket aces you need to use a different kind of probability.

Before teaching you about this type of probability you need to make sure you want to try to learn it. If you’re getting confused or worried about all of this math, skip to the pot odds section. It builds on what you’ve already learned and you don’t have to know anything about this next part in order to be a master Texas holdem player.

The main difference is you need to look at the possibilities of how many events out of how many possibilities instead of the positive verses the negative. This is easier to understand using an example.

As you’ve already seen when you determine a ratio or odds you compare the cards that can help you and the ones that can’t. When you determine the possibility of getting dealt pocket aces you need to use the number of cards that can help achieve this out of the total number of cards.

So the odds of getting an ace as the first card are 4 out of 52 and as your second card are 3 out of 51. After the first card there are only 51 remaining cards in the deck and only 3 remaining aces.

Here’s why this distinction is important. To get the actual chances of being dealt pocket aces you put 4 over 52, like a fraction, and 3 over 51 and multiply them.

This gives you the chances of being dealt pocket aces as 220 to 1.

When you multiply the fractions you get 12 over 2,652, which reduce to 1 over 221. This is then made back into a ratio like discussed above. 1 time you receive pocket aces and 220 times you don’t, so the odds are 220 to 1.

Outs

Once you understand how odds work, the next thing you must understand is how to determine how many outs you have in any situation where you need to understand your pot odds.

You combine your knowledge of odds with the number of outs and the amount in the pot and of the bet you’re facing to figure out your pot odds.

Your outs are the cards that improve your hand enough to win the pot. Rarely can you be 100% sure about every one of your outs, but in most situations you can make an educated guess.

Example

If you have a king and a ten and the flop has an ace and a king, if one of your opponents bets the odds are they have an ace. This means you’re behind in the hand, but if you hit one of the other two kings or one of the remaining three tens on the turn or river you stand a good chance to win.

You can’t be certain either a king or ten will win the hand because your opponent may have a set of aces or two pair with aces and either kings or the other flop card. This isn’t likely but it does happen.

One of the most common situations concerning outs is when you have four to a flush on the flop. This leaves nine outs to complete the flush. Each suit has 13 cards and you have four of them, leaving nine outs.

But will each of the nine cards guarantee a win?

Rarely is anything guaranteed at the holdem table. If the board pairs and you hit your flush you can still lose to a full house. If you don’t have an ace high flush you might lose to a higher flush.

When you’re trying to determine how many outs you have, try to make a realistic guess on how many will actually win the hand.

You need to use everything that has happened in the hand so far, what you know about your opponents, and the range of hands your opponent is likely holding to make your best estimate.

This information is directly related to the level of competition you’re facing and the limits of the table.

Example

When you’re playing at micro or low limits and your competition isn’t very good overall it’s likely that any flop that holds an ace has paired an ace in someone’s hand. Low limit players often play any ace. While better players and higher limit players also play hands with aces, they usually don’t play the ones with smaller side cards like you see at the lower levels.

When you’re trying to determine your number of outs you can adjust the number based on different factors. If you have 10 outs but think that 10% of the time your opponent will hold a better hand anyway you can adjust your outs to nine when using it for your pot odds calculation.

This can get quite complicated, but as you gain experience and get better at reducing the range of possible hands your opponent holds your outs determination will improve.

If you knew exactly what your opponent held you could determine your exact number of outs every time.

To complicate your outs computation further, the more opponents who remain in the hand the more difficult it can be to make an accurate outs guess.

If you’re in a hand with an ace and a seven and three other opponents and an ace lands on the flop do you have the best hand? If not how likely is it you can improve to win the hand?

In many games you need to be worried that one of your opponents also has an ace and her kicker is likely better than yours. You probably need to make two pair to have a chance to win, but what are the odds she’ll also hit two pair? One of your opponents may already have two pair or a set.

In a no limit game this hand is rarely profitable, because if you do hit two pair the only way you usually get enough action to make it possibly profitable is when another player has a better hand. It can be profitable in a few limit games, but usually a hand like this needs to be folded before the flop to keep from getting into an uncertain situation like the one I described.

It’s impossible to cover every situation you’ll run into concerning outs. Do the best you can to determine your real outs and learn to adjust your counts as you gain experience. You can use the examples of outs in the examples section below to practice and learn how to think about each situation.

Pot Odds

Pot odds are what Texas holdem players use to determine if calling or folding is the correct course of action when facing a bet. They’re mostly used on the flop, turn, and river.

While it’s possible to consider pot odds before the flop, most players focus on their position and starting hand strength for most of their pre flop decisions. This is the correct way to play and more profitable than thinking about pre flop pot odds so this section deals almost entirely with post flop play.

Once you master every other aspect of pre flop play you can consider thinking about pot odds before the flop, but only rarely will it prove profitable.

Of course you need to learn how to correctly use pot odds in both limit and no limit play, but the easiest way for a beginner to start learning about them is with a simple limit game example. Considerations for limit and no limit play are covered in a section later on this page.

Example

You’re playing in a $2 / $4 limit Texas holdem game and the blinds are $1 and $2. Two players before you call for $2 each, you call, and the small blind puts another dollar in the pot and the big blind checks. This leaves a total of $10 in the pot.

You have the ace of spades and eight of spades and the flop is the king of clubs, jack of spades, and six of spades. This gives you four to the nut flush, so you have nine outs out of 47 unseen cards. 38 cards won’t help and 9 will complete the flush for a ratio of 38:9 to complete the flush.

It’s also possible you already have the best hand, but this isn’t likely with four opponents. You may also win the hand if you hit one of the remaining aces.

This is a perfect example of one of the situations mentioned in the outs section above. Unless the board pairs, if you hit a flush you’re guaranteed to win the hand. Even if the board pairs, you’ll win most of the time. You’ll win some of the time when you hit an ace, but not all of the time.

In this situation the safest way to play is ignore everything except the flush draw at this point for your outs. The closest to true outs you probably have are 10. The 9 flush outs and an additional out for the times you win when an ace hits.

The problem is you don’t know when an ace helps you and when it simply costs you more money.

Let’ continue with the example using the 9 outs for the flush.

The small blind bets $2 into the pot, two players fold and one other player calls, leaving a total of $14 in the pot. You have to call the bet of $2 so the pot is offering odds of 14 to 2, or 7 to 1. This means if your odds of hitting your hand are better than 7 to 1 it’s the correct play mathematically to call.

Your odds are 38 to 9, or roughly 4.2 to 1. This means you have a much better chance to hit your flush than it costs you to call. You’ll still only hit the flush 1 out of every 5 times you’re in this situation, but the amount you win for the call of $2 offers 7 times the investment, making it clearly profitable in the long run.

What if the first player bet $2 and the other remaining player raised to $4?

This makes the total in the pot $16 and you have to call a bet of $4, giving you pot odds of 4 to 1. Your odds of hitting your hand are still 4.2 to 1, so now it shows a call may be slightly less than break even in the long run.

But don’t forget the player that made the original bet of $2. She’ll probably call the raise enough times to make it worth calling. If you know she’s going to call the pot is offering 18 to 4, or 4.5 to 1 pot odds.

One of the big problems with a hand like a flush draw is you rarely get a great deal of action after you hit your hand. Some hands are more concealed and offer the chance to win more when you hit your hand. This is covered in the implied odds section below.

So far we’ve only covered situations immediately following the flop. Once the turn card is dealt you have to do the same thing for the river to determine the best course of action using pot odds.

Of course you have both the turn and river to hit your outs, but when you determine your pot odds you should only consider the turn or try to correctly guess how much it’ll cost to play both the turn and river and the amount of the total pot including both bets and the bets of your opponents.

You can learn how to do this fairly accurately and it involves implied odds, but until you master pot odds and outs you should focus on each section of the hand by itself. Focus on becoming the best turn player possible, then the best river player possible before trying to combine the two.

What about the money you put in the pot before the flop?

In the example above you put $2 in the pot before the flop, so why don’t you factor it in your calculation? The truth is you’re using it in your calculations already, but it’s already in the pot so it’s not yours any more. The only way you get it back is by winning.

Shortcuts

One of the fastest ways to learn to use pot odds is to memorize how many outs you have in certain situations. Here’s a list of common situations and how many outs they have.

• A flush draw has 9 outs.
• An open end straight draw has 8 outs.
• An inside straight draw has 4 outs.
• You have 6 outs to pair one of your hole cards unless they’re a pair.
• A pocket pair has 2 outs to hit a set.
• A set has 7 outs on the turn to hit quads or a full house and 10 on the river.

Another shortcut that many players find useful also involves the number of outs. Once you determine how many outs you have you can multiply it by two to get an idea of the percentage chance you have of hitting your hand on the turn. You can also multiply by two after the turn to get an idea of your chances on the river.

You can multiply by four to determine your chances on the turn and river combined.

Example

If you have four to a flush on the flop you have roughly an 18% chance to hit your flush on the turn and a 36% chance on either the turn or river. Nine outs time two is 18 and times four is 36.

Notice that the statement was you could get a close idea or general idea. These are estimates, not exact numbers.

The actual percentage of hitting the flush on the turn is 19.1%, not 18% and the exact chance on the turn or river is 35%, not 36%.

You have to have at least 14 outs after the flop in order to be the favorite to hit your hand on the turn or river. With 14 outs the exact percentage is 51.2%

Poker Hands Probability Explained Rules

Here’s a chart with the percentages based on the number of your outs on the turn and on the turn and river and how these percentages relate to the odds.

Outs	Turn	Odds	Turn and River	Odds
20	42.6%	1.4 to 1	67.5%	.48 to 1
19	40.4%	1.5 to 1	65%	.54 to 1
18	38.3%	1.6 to 1	62.4%	.6 to 1
17	36.2%	1.8 to 1	59.8%	.67 to 1
16	34%	1.9 to 1	57%	.75 to 1
15	31.9%	2.1 to 1	54.1%	.85 to 1
14	29.8%	2.2 to 1	51.2%	.95 to 1
13	27.7%	2.6 to 1	48.1%	1.1 to 1
12	25.5%	2.9 to 1	45%	1.2 to 1
11	23.4%	3.3 to 1	41.7%	1.4 to 1
10	21.3%	3.7 to 1	38.4%	1.6 to 1
9	19.1%	4.2 to 1	35%	1.9 to 1
8	17%	4.9 to 1	31.5%	2.2 to 1
7	14.9%	5.7 to 1	37.8%	2.6 to 1
6	12.8%	6.8 to 1	24.1%	3.2 to 1
5	10.6%	8.4 to 1	20.3%	3.9 to 1
4	8.5%	10.8 to 1	16.5%	5.1 to 1
3	6.4%	14.7 to 1	12.5%	7 to 1
2	4.3%	22.5 to 1	8.4%	10.9 to 1
1	2.1%	46 to 1	4.3%	22.3 to 1

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It’s important to understand how to use this information in relation to pot odds like you learned above.

The most common example is if you face an all in bet on the flop. If you ignore everything else like the size of the pot, if you have 14 or more outs you’re favored to win the hand.

If the pot has $200 in it before your opponent bets and the all in bet is $100, you’re receiving 3 to 1 pot odds so if you have seven or more outs you need to call.

You can print this table and keep it with you as you play online. As you look up different situations you’ll start remembering the outs and odds and soon you won’t need to use the sheet to make the best plays.

Limit vs No Limit Play

Though many players act like there’s a huge difference between limit and no limit play when it comes to pot odds, the truth is everything is the same.

The only difference is the size of the bets, but that doesn’t change the way you determine the outs, odds, and pot odds. It only changes the amount in the pot and the amount of the possible bets.

Some situations are so close that it’s hard to tell if calling or folding is the correct play based on the pot odds. When this happens it’s often helpful to consider the possibilities of the size of the pot during the rest of the hand. You’ll learn more about this in the next section.

Implied Odds

Implied odds are the odds of winning more than the current bet when you hit your hand.

When you hit your hand on the turn you have additional opportunities to win money from your opponent. If you miss your hand on the river you can fold, but you might be able to win an additional bet from your opponent when you hit your hand.

When you start trying to figure out implied odds you need to be able to guess how likely it is that your opponent will make or call additional bets when you hit your hand.

The more disguised your hand is the more likely your opponent will pay you off.

When you’re drawing to a flush it’s usually obvious to your opponent that you may have completed it because three cards of the same suit are on the board. But when you hit a straight or a set it’s usually more difficult for your opponent to see what you hit.

If your opponent understands pot odds the good news is by the time you hit your flush the pot will be large enough that she should call most bets. But when you hit a straight or other hidden hand you’ll be able to get your opponent to call additional bets most of the time.

In limit play most players call a single bet on the river even if they’re sure they’re beat. It’s a little more complicated in no limit. The key is to learn how to size your bets to make your opponent call when you hit your hand and be able to avoid the same thing happening to you when you miss your draw.

Here’s an example.

You have a suited ace and queen, paired the queen on a king high flop, and missed the flush draw on the river. The pot has $1,000 in it and you’ve been calling an aggressive player the entire hand. You only have a pair of queens and the odds are strongly against you having the best hand.

How big of a bet will you call on the river?

Believe it or not, this question has a mathematical answer, but you have to guess how often you’ll have the best hand.

Roughly, if you think you have a 10% chance of winning the hand you’d call a $100 bet.

The way you determine the correct amount based on your chances of winning is by putting yourself in the situation 100 times.

• In the above example it costs $10,000 to call the $100 bet 100 times.
• If you win 10% of the time you win 10 times and lose 90 times.
• The 10 times you win you win $12,000 for a profit of $2,000.

Realize that the $12,000 includes the money you use to call the bet. The reason you have to include it is because you use it to determine the $10,000 needed to call. The actual percentage you need to win is only 9% of the time to show a profit, but using 10% is much easier to work with in your head.

You only have to win 25% of the time to call a $500 bet to break even. You can figure this the same way as you did the 10% above, but let’s look at it a different way. The $500 bet makes the pot $1,500 and you have to call $500. This makes the pot odds 3 to 1 so you only need to win 1 out of every 4 times, or 25%, to break even.

The truth is that when you get enough experience you won’t really use the math as much as your instinct and what you know about your opponent. You have to call occasionally in this situation to show you can’t be pushed around, but you can’t afford to call a very big wager and be wrong very often either.

Example

The easiest way to learn about odds, outs, and pot odds for most people is to read how to determine them and then practice them. This section is filled with examples split into three areas.

You’ll find real life examples of odds, outs, and pot odds. Each one has a situation for you to practice what you’ve just learned.

The situations are listed first, and then the solutions are included below so you won’t see the answers until you scroll down the page.

Odds

Your first card is an ace, what are the odds your second card will be an ace?

The deck has 51 cards remaining and 3 of them are aces, so 3 out of 51 will be an ace. As a ration, 3 cards help you and 48 don’t.

Your first card is the ace of spades. What are the odds your second card will be a spade?

Of the remaining 51 cards, 12 are spades so 12 out of 51 will be a spade. As a ration 12 help you and 39 don’t.

You have a set of eights after the flop. What are the odds the turn will be the last eight?

You have 47 unseen cards and only 1 eight so 1 out of 47 will be the last eight. As a ratio 1 card will help and 46 won’t.

You have a set of eights after the turn. What are the odds the river will be the last eight?

Now there are only 46 unseen cards so 1 out of 46 will help and as a ratio 1 will help and 46 won’t.

Now there are only 46 unseen cards so 1 out of 46 will help and as a ratio 1 will help and 46 won’t.

With 47 unseen cards, you can pair either of the two community cards that aren’t part of your set, so you have 6 outs. 6 out of 47 will make a full house so as a ratio you have 6 that help and 41 that don’t.

You have a set of eights after the turn. What are the odds the river will make a full house?

Poker Hands Probability Explained Answers

Now you have 46 unseen cards but have picked up another card to pair. So you have 9 outs out of 46 cards. As a ratio you have 9 cards that complete a full house and 37 that don’t. Remember that one of the cards that don’t make a full house gives you quads, so you actually have 10 outs instead of 9.

Outs

You have a set on the flop. How many outs do you have to improve to a full house or quads?

Poker Hands Probability Explained Probability

You have one out to improve to quads. A set dictates that you have a pocket pair and one of the board cards matches it. So the flop also contains two unpaired cards, so you have an additional six outs from these. If they were paired on the board you’d already have a full house. If you don’t hit a full house on the turn your outs improve by three for the river. The turn card can also pair on the river for a full house. So you have seven outs on the turn and 10 on the river.

You have four to a flush on the flop. How many outs do you have to complete your flush?

Four to a flush leaves nine more cards of the needed suit. Each suit has 13 cards, minus the four in your hand and on the board.

You have four to a flush and two over cards to the flop. How many outs do you have assuming if you pair one of your cards you’ll win?

In addition to the nine outs from the flush, each of your over cards has three more cards that can match them. This provides a total of 15 outs.

You have an open end straight draw. How many outs do you have?

An open end straight draw has two cards that can complete the straight and each card has four in the deck. This gives a total of eight outs.

You have an open end straight draw and one over card. How many outs do you have assuming your over card will win if you pair it?

The single over card adds three more outs to the open end straight draw so the total number of outs is 11.

You have an inside straight draw. How many outs do you have?

An inside straight draw only has one card that can complete the straight so you have four outs.

You have two over cards. How many outs do you have if pairing either one wins?

Each over car has three remaining cards in the deck, so a total of six outs remain.

You have a flush on the flop. How many outs does your opponent have if they currently have two pair?

Your opponent has to make a full house in order to beat your flush. With two pair she has two remaining cards for each of her pairs. This makes a total of four outs.

You have an open end straight draw and four to a flush. How many outs do you have?

A flush draw has nine outs and an open end straight draw has eight outs, but a few of the outs may overlap. For example, if you have the eight and nine of hearts and the board has the seven of hearts, six of diamonds, and three of hearts, any of the remaining hearts will give you a flush and any of the tens or fives will complete your straight, But you’ve already counted the ten of hearts and the five of hearts in your nine outs so you only have six additional open end straight outs instead of eight. This is a total of 15 outs.

You have two over cards, four to a flush, and an open end straight draw. How many outs do you have? Example: You have king and queen of clubs and the board is jack of clubs, ten of diamonds, and seven of clubs.

You just determined how many outs a flush draw and open end straight draw include, so you just need to add the two over cards. Each of them has three remaining cards, so a total of 21 outs. Take a quick look at the chart included above. With 21 outs you’re a strong favorite to win the hand, even if you’re currently behind. Of course it’s possible not all of your 21 outs are actually good.

Pot Odds

The pot has $100 in it after the flop including a bet of $20. You’re the only other player and you have a flush draw. Should you call based on your pot odds?

A flush draw has 9 outs so the odds are 4.2 to 1. The pot is offering 5 to 1 odds. % to 1 is better than 4.2 to 1 so you should call.

The pot has $40 in it after the flop including a bet of $20. You’re the only player left to act and you have a flush draw. Should you call?

The flush draw has 4.2 to 1 odds and the pot is only offering 2 to 1. You should fold because 2 to 1 is lower than 4.2 to 1. This means in the long run you’ll lose money by calling.

You have an open ended straight draw and four to a flush on the flop, you’re facing a bet of $100 and the pot size is $400. What should you do?

You have 15 outs so the odds are 2.13 to 1 to complete one of your draws on the turn, but on the turn and river the odds are .85 to 1. This means you’re actually a favorite to win the hand so it doesn’t matter what the pot odds are, you should raise. The pot odds are 4 to 1.

You have a set on the flop but the flop is all the same suit and it looks like one of your two remaining opponents may have a flush. The pot size is $320 including a bet and call of $40 each, so you’re the last to act on the round. What should you do?

You only have 6 outs to make a full house on the flop so the odds are 6.8 to 1 and the pot is offering 8 to 1. You should call. Even if the odds were as low as 6 to 1 you should consider calling because even if the turn doesn’t complete your full house the odds improve to 4.1 to 1 on the river. This is a fine line and represents a answer that shows why pot odds are so important. Knowing what happens on the next round as far as odds, outs, and pot odds, gives you the ability to make the correct play on close answers.

The river was just dealt, the pot has $1,000 in it including a bet of $200 and you have two pair. You’re not sure what your opponent has. How often do you need to have the winning hand to make a call correct?

If you’re in this answer 100 times it costs a total of $20,000 and when you win you receive $1,200 including your call. You only have to win 17 times out of 100, or 17% of the time, in order to show a long term profit. This means the correct play is to call, because the odds of you not being good at least 17% of the time is small.

Some Final Words

No matter how hard it seems at first, learning how to use odds, pot odds, and outs is worth it in the long run. If you’re having trouble keeping everything straight, go over the qui part of this page every day and print the chart and start using it.

Poker Hands Probability Explained Calculator

Once you learn how to use the things you’ve learned on this page at the Texas holdem tables you’ll quickly see an improvement in your profitability.