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 be confident the numbers are correct. Here is a table summarizing the number of 7-card poker hands. The probability is the probability of having the hand dealt to you when dealt 7 cards. 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. How to Compute Poker Probabilities. In a previous lesson, we explained how to compute probability for any type of poker hand. For convenience, here is a brief review: Count the number of possible five-card hands that can be dealt from a standard deck of 52 cards; Count the number of ways that a particular type of poker hand can occur. We determine the number of 6-card poker hands. There are a few 6-card poker games so it is worth looking at probabilities for winning with certain kinds of hands. One chooses the highest ranked 5-card poker hand among the 6 cards and values the hand based on the 5-card hand. The types of 5-card poker hands in decreasing rank are.

Brian Alspach

17 January 2000

Abstract:

There are a few 6-card poker games so it is worth looking at probabilitiesfor winning with certain kinds of hands. One chooses the highest ranked5-card poker hand among the 6 cards and values the hand based on the5-card hand. The types of 5-card poker hands in decreasing rank 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 6-card poker hands is .

A straight flush is completely determined once the smallest card in thestraight flush is known. There are 40 cards eligible to be the smallestcard in a straight flush. If the smallest card in the straight flush isan ace, then the sixth card may be any of 47 cards. If the smallest cardin the straight flush is any of the other 36 eligible beginning cards,then the sixth card may be any of 46 cards because we cannot use the nextsmallest card in the same suit as the straight flush. Hence, there are6-card hands containing straight flushes.

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 2 cards. This implies there are 4-of-a-kind hands.

There are 2 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. There are ways to choose the 2 ranksand 4 ways to choose each of the triples. This gives us full houses of this type. The other way of getting a full houseis for the 6-card hand to contain a triple, a pair and a remaining card ofa third rank. There are 13 choices for the rank of the triple, 12 choicesfor the rank of the pair, and 44 choices for the singleton card. Thereare 4 ways of choosing the triple of a given rank and 6 ways to choose thepair of the other rank. This produces full houses of the latter type. Adding the two numbersyields 165,984 full houses.

To count the number of flushes, we obtain 6-card hands formed from cards in the same suit. Altogether, thereare flushes with 6 cards in the same suit.There are choices for 5 cards in the same suit.There are then 39 choices for a sixth card from a different suit. Thus,there are flushes in 6-card hands, whereprecisely 5 cards are in the same suit. Combining the two gives us207,636 6-card hands containing flushes. Of these, 1,844 are straightflushes whose removal leaves 205,792 flushes.

Let's determine how many sets of 6 distinct ranks correspond to straights.One possible form is ,where x can be any of9 ranks. The other possible form is ,where yis neither x-1 nor x+5. When x is ace or 10, then there are 7choices for y. When x is between 2 and 9, inclusive, there are 6choices for y. This implies there are sets of 6 distinct ranks corresponding to straights. There are then 4choices for each card of the given ranks except we must remove thosechoices producing flushes. There are 4 choices giving all 6 cards inthe same suit. If 5 are in the same suit, there are choices of which 5 ranks will be in the same suit, 4 choicesfor the suit of the 5 cards, and 3 choices for the suit of the remainingcard. So there are choices which give aflush. This means there are 4⁶ - 76 = 4,020 choices not producing aflush. Hence, there are straights of thisform.

We also can have a set of 5 distinct ranks producing a straight whichmeans the corresponding 6-card hand must contain a pair as well. Thereare 10 sets of ranks of the form .There are 5choices of the rank to be paired, 6 choices for the pair, and 4 choicesfor each of the other 4 cards except not all 4 cards can be chosen inthe same suit as either of the cards in the pair. This means there are4⁴ - 2 = 254 choices for the 4 cards. We then have straights of this form. Altogether there are361,620 straights.

In forming a 3-of-a-kind hand, there must be a triple and 3 other cardsall of distinct ranks different from the rank of the triple. There are13 choices for the rank of the triple, and there are choices for the ranks of the other 3 cards. There are 4 choicesfor the triple of the given rank and there are 4 choices for each of thecards of the remaining 3 ranks. Altogether, we have 3-of-a-kind hands.

Next we consider two pairs hands. Such a hand may contain either threepairs, or two pairs plus two remaining cards of distinct ranks. Weevaluate these 2 types of hands separately. If the hand has threepairs, there are ways to choose the ranks ofthe pairs and 6 ways to choose each of the pairs. This produces6-card hands with three pairs.

For the other kind of two pairs hand, there are choices for the two ranks of the pairs and there are ways to choose the ranks of the 2 singletons. There are 6 choicesfor each of the pairs, and there are 4 choices for each of the remaining2 cards. This produces hands of two pairs of the second type. Adding the two gives 2,532,8166-card hands with two pairs.

Now we count the number of hands with a pair. Such a hand must have5 distinct ranks. There are possible setsof 5 ranks. We must remove sets of the form because these correspond to straights. There are 10 such sets leaving1,277 sets of ranks corresponding to a hand with one pair. Given sucha set, there are 5 choices for the rank of the pair, and 6 choices fora pair of the chosen rank. There are 4 choices for each of the remaining4 cards except we cannot choose all 4 to be in the same suit as eitherof the cards forming the pair. Hence, there are 4⁴-2 = 254 choicesfor 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 20,358,520 which will serve as a check on our arithmetic.

A high card hand has 6 distinct ranks, but does not include straights.So we must eliminate sets of ranks which have 5 consecutive ranks. Indetermining the number of straights above, we derived that there are 71sets of 6 distinct ranks which give straights. There are sets of 6 distinct ranks. Removing the 71 sets correspondingto straights, leaves 1,645 sets of distinct ranks which do not producestraights. There are 4 choices for each of the 6 cards in a given setproducing 4⁶ = 4,096 ways of choosing cards for a given set of ranks.However, some of the choices produce flushes and we must remove them.Clearly there are 4 ways of choosing the 6 cards all in the same suitwhich is one way of getting a flush. There 6 ways of choosing 5 of theranks and 4 choices for the suit of these 5 ranks, and 3 choices for thesuit of the remaining card. This gives us choices of suits which produce a flush with 5 cards in the same suit.We remove these 76 choices which produce flushes giving us 4,020 choicesfor the 6 cards which do not produce a flush. Multiplying 4,020 by 1,645gives 6,612,900 high card hands.

If we sum the preceding numbers, we obtain 20,358,520 and we can be confidentthe numbers are correct.

Here is a table summarizing the number of 6-card poker hands. Theprobability is the probability of having the hand dealt to you whendealt 6 cards.

Poker Hand Probabilities Holdem

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 6-card hands based on 6 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 a6-card hand with a single pair and call it a high card hand. Thiswould have the effect of creating the following distortion. Thereare 16,343,640 6-card hands containing 5 cards which are high cardhands. Of these 16,343,640 hands, 9,730,740 also contain 5-cardhands which have a pair. Thus, the latter hands are more specialand should be ranked higher (as they indeed are) but would not beunder the scheme being discussed in this paragraph.

Want to find out the probabilities of being dealt an ace before the flop in Texas Hold'em? Use the tables below to help you find the different aces odds based on the situation and the number of players at the table.

Aces probabilities odds charts.

Probability of being dealt an ace in any Texas Hold'em game is 15%.

How to use aces odds.

These tables are not going to be of much use to you whilst you play, but it is useful information on the probabilities of hands with aces turning up at your table.

The most useful information from these aces odds tables is that the likelihood of a player being dealt an ace increases with the number of players at the table.

The likelihood of another player holding an ace if you hold an ace yourself table (top right) is interesting. There is a fairly high probability that another player at a 9-seater table holds an ace if you do as well. This means that we should be cautious when dealt weak aces and should fold them in early position. There is a chance that we were the only player at the table holding an ace, but we are saving ourselves more money in the long run for when we come up against another player with a better ace than ours.

Playing aces in Texas Hold'em.

An ace is the highest ranking card in Texas Hold'em and can be played either high or low. This means that if you are dealt an ace preflop, there is a fair chance that you hold one of the strongest hands at the table. If you are all-in with an opponent and they do not have a pocket pair or a better ace, you are the favorite to win the hand.

However, you should be careful to not become overly attached to the hand when you are dealt an Ace, because there is a fair chance that you will lose a sizeable proportion of your stack if you get into a hand with another player that holds a better ace than you (e.g. A9 vs AQ).

Be careful to not overvalue hands with aces, as they can get you in to a lot of trouble if you do not have a good kicker.

For strategy guides on how to play hands with aces, take a look at the rag aces and pocket aces articles.

Go back to the poker odds charts.

Poker Hand Probabilities Math

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Poker Hands And Probabilities Symbols

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