Poker Run Rules and Tally Sheet Present your Tally Sheet, then pick one card at a time, and lay it face up on the table. No duplicate cards can be used. Participants must draw an additional card if an identical card is drawn. If playing MORE than one hand, the participant must present the appropriate Poker Hand. Poker Odds Calculator Odds of Winning w/ Any Poker Hand PokerListings.com’s Poker Odds Calculator is the fastest, most accurate and easy-to-use poker odds calculator online. All players play their 6 7 8 9 10 poker hand best five cards using any combination of their two hole cards and the five community cards on the table. Even with a conservative press schedule, it would have left hundreds on the table.

1. Poker Hand 7 8 9 10 Q
2. 6 7 8 9 10 Poker Hand

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Introduction

Derivations for Five Card Stud

I have been asked so many times how I derived the probabilities of drawing each poker hand that I have created this section to explain the calculation. This assumes some level mathematical proficiency; anyone comfortable with high school math should be able to work through this explanation. The skills used here can be applied to a wide range of probability problems.

The Factorial Function

If you already know about the factorial function you can skip ahead. If you think 5! means to yell the number five then keep reading.

The instructions for your living room couch will probably recommend that you rearrange the cushions on a regular basis. Let's assume your couch has four cushions. How many combinations can you arrange them in? The answer is 4!, or 24. There are obviously 4 positions to put the first cushion, then there will be 3 positions left to put the second, 2 positions for the third, and only 1 for the last one, or 4*3*2*1 = 24. If you had n cushions there would be n*(n-1)*(n-2)* ... * 1 = n! ways to arrange them. Any scientific calculator should have a factorial button, usually denoted as x!, and the fact(x) function in Excel will give the factorial of x. The total number of ways to arrange 52 cards would be 52! = 8.065818 * 10⁶⁷.

The Combinatorial Function

Assume you want to form a committee of 4 people out of a pool of 10 people in your office. How many different combinations of people are there to choose from? The answer is 10!/(4!*(10-4)!) = 210. The general case is if you have to form a committee of y people out of a pool of x then there are x!/(y!*(x-y)!) combinations to choose from. Why? For the example given there would be 10! = 3,628,800 ways to put the 10 people in your office in order. You could consider the first four as the committee and the other six as the lucky ones. However you don't have to establish an order of the people in the committee or those who aren't in the committee. There are 4! = 24 ways to arrange the people in the committee and 6! = 720 ways to arrange the others. By dividing 10! by the product of 4! and 6! you will divide out the order of people in an out of the committee and be left with only the number of combinations, specifically (1*2*3*4*5*6*7*8*9*10)/((1*2*3*4)*(1*2*3*4*5*6)) = 210. The combin(x,y) function in Excel will tell you the number of ways you can arrange a group of y out of x.

Now we can determine the number of possible five card hands out of a 52 card deck. The answer is combin(52,5), or 52!/(5!*47!) = 2,598,960. If you're doing this by hand because your calculator doesn't have a factorial button and you don't have a copy of Excel, then realize that all the factors of 47! cancel out those in 52! leaving (52*51*50*49*48)/(1*2*3*4*5). The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability.

Poker Math

The next section shows how to derive the number of combinations of each poker hand in five card stud.

Royal Flush

There are four different ways to draw a royal flush (one for each suit).

Straight Flush

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight flushes.

Four of a Kind

There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 48. Thus there are 13 * 48 = 624 different four of a kinds.

Full House

There are 13 different possible ranks for the three of a kind, and 12 left for the two of a kind. There are 4 ways to arrange three cards of one rank (4 different cards to leave out), and combin(4,2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a full house.

Flush

There are 4 suits to choose from and combin(13,5) = 1,287 ways to arrange five cards in the same suit. From 1,287 subtract 10 for the ten high cards that can lead a straight, resulting in a straight flush, leaving 1,277. Then multiply for 4 for the four suits, resulting in 5,108 ways to form a flush.

Straight

The highest card in a straight can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible high cards. Each card may be of four different suits. The number of ways to arrange five cards of four different suits is 4⁵ = 1024. Next subtract 4 from 1024 for the four ways to form a flush, resulting in a straight flush, leaving 1020. The total number of ways to form a straight is 10*1020=10,200.

Three of a Kind

There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are combin(12,2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus the number of ways to arrange a three of a kind is 13 * 4 * 66 * 4² = 54,912.

Two Pair

There are (13:2) = 78 ways to arrange the two ranks represented. In both ranks there are (4:2) = 6 ways to arrange two cards. There are 44 cards left for the fifth card. Thus there are 78 * 6² * 44 = 123,552 ways to arrange a two pair.

One Pair

There are 13 ranks to choose from for the pair and combin(4,2) = 6 ways to arrange the two cards in the pair. There are combin(12,3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 4³ = 1,098,240 ways to arrange a pair.

Nothing

First find the number of ways to choose five different ranks out of 13, which is combin(13,5) = 1287. Then subtract 10 for the 10 different high cards that can lead a straight, leaving you with 1277. Each card can be of 1 of 4 suits so there are 4⁵=1024 different ways to arrange the suits in each of the 1277 combinations. However we must subtract 4 from the 1024 for the four ways to form a flush, leaving 1020. So the final number of ways to arrange a high card hand is 1277*1020=1,302,540.

Specific High Card

For example, let's find the probability of drawing a jack-high. There must be four different cards in the hand all less than a jack, of which there are 9 to choose from. The number of ways to arrange 4 ranks out of 9 is combin(9,4) = 126. We must then subtract 1 for the 10-9-8-7 combination which would form a straight, leaving 125. From above we know there are 1020 ways to arrange the suits. Multiplying 125 by 1020 yields 127,500 which the number of ways to form a jack-high hand. For ace-high remember to subtract 2 rather than 1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and 5-4-3-2-A are both valid straights. Here is a good site that also explains how to calculate poker probabilities.

Five Card Draw — High Card Hands

Hand	Combinations	Probability
Ace high	502,860	0.19341583
King high	335,580	0.12912088
Queen high	213,180	0.08202512
Jack high	127,500	0.04905808
10 high	70,380	0.02708006
9 high	34,680	0.01334380
8 high	14,280	0.00549451
7 high	4,080	0.00156986
Total	1,302,540	0.501177394

Ace/King High

For the benefit of those interested in Caribbean Stud Poker I will calculate the probability of drawing ace high with a second highest card of a king. The other three cards must all be different and range in rank from queen to two. The number of ways to arrange 3 out of 11 ranks is (11:3) = 165. Subtract one for Q-J-10, which would form a straight, and you are left with 164 combinations. As above there 1020 ways to arrange the suits and avoid a flush. The final number of ways to arrange ace/king is 164*1020=167,280.

Internal Links

For lots of other probabilities in poker, please see my section on Probabilities in Poker.

Poker Hand 7 8 9 10 Q

This page describes the ranking of poker hands. This applies not only in the game of poker itself, but also in certain other card games such as Chinese Poker, Chicago, Poker Menteur and Pai Gow Poker.

• Low Poker Ranking: A-5, 2-7, A-6
• Hand probabilities and multiple decks - probability tables

Standard Poker Hand Ranking

There are 52 cards in the pack, and the ranking of the individual cards, from high to low, is ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. In standard poker - that is to say in the formal casino and tournament game played internationally and the home game as normally played in North America - there is no ranking between the suits for the purpose of comparing hands - so for example the king of hearts and the king of spades are equal. (Note however that suit ranking is sometimes used for other purposes such as allocating seats, deciding who bets first, and allocating the odd chip when splitting a pot that can't be equally divided. See ranking of suits for details.)

A poker hand consists of five cards. The categories of hand, from highest to lowest, are listed below. Any hand in a higher category beats any hand in a lower category (so for example any three of a kind beats any two pairs). Between hands in the same category the rank of the individual cards decides which is better, as described in more detail below.

In games where a player has more than five cards and selects five to form a poker hand, the remaining cards do not play any part in the ranking. Poker ranks are always based on five cards only, and if these cards are equal the hands are equal, irrespective of the ranks of any unused cards.

Some readers may wonder why one would ever need to compare (say) two threes of a kind of equal rank. This obviously cannot arise in basic draw poker, but such comparisons are needed in poker games using shared (community) cards, such as Texas Hold'em, in poker games with wild cards, and in other card games using poker combinations.

1. Straight Flush

If there are no wild cards, this is the highest type of poker hand: five cards of the same suit in sequence - such as J-10-9-8-7. Between two straight flushes, the one containing the higher top card is higher. An ace can be counted as low, so 5-4-3-2-A is a straight flush, but its top card is the five, not the ace, so it is the lowest type of straight flush. The highest type of straight flush, A-K-Q-J-10 of a suit, is known as a Royal Flush. The cards in a straight flush cannot 'turn the corner': 4-3-2-A-K is not valid.

2. Four of a kind

Four cards of the same rank - such as four queens. The fifth card, known as the kicker, can be anything. This combination is sometimes known as 'quads', and in some parts of Europe it is called a 'poker', though this term for it is unknown in English. Between two fours of a kind, the one with the higher set of four cards is higher - so 3-3-3-3-A is beaten by 4-4-4-4-2. If two or more players have four of a kind of the same rank, the rank of the kicker decides. For example in Texas Hold'em with J-J-J-J-9 on the table (available to all players), a player holding K-7 beats a player holding Q-10 since the king beats the queen. If one player holds 8-2 and another holds 6-5 they split the pot, since the 9 kicker makes the best hand for both of them. If one player holds A-2 and another holds A-K they also split the pot because both have an ace kicker.

3. Full House

This combination, sometimes known as a boat, consists of three cards of one rank and two cards of another rank - for example three sevens and two tens (colloquially known as 'sevens full of tens' or 'sevens on tens'). When comparing full houses, the rank of the three cards determines which is higher. For example 9-9-9-4-4 beats 8-8-8-A-A. If the threes of a kind are equal, the rank of the pairs decides.

4. Flush

Five cards of the same suit. When comparing two flushes, the highest card determines which is higher. If the highest cards are equal then the second highest card is compared; if those are equal too, then the third highest card, and so on. For example K-J-9-3-2 beats K-J-7-6-5 because the nine beats the seven.If all five cards are equal, the flushes are equal.

5. Straight

Five cards of mixed suits in sequence - for example Q-J-10-9-8. When comparing two sequences, the one with the higher ranking top card is better. Ace can count high or low in a straight, but not both at once, so A-K-Q-J-10 and 5-4-3-2-A are valid straights, but 2-A-K-Q-J is not. 5-4-3-2-A, known as a wheel, is the lowest kind of straight, the top card being the five.

6. Three of a Kind

Three cards of the same rank plus two unequal cards. This combination is also known as Triplets or Trips. When comparing two threes of a kind the rank of the three equal cards determines which is higher. If the sets of three are of equal rank, then the higher of the two remaining cards in each hand are compared, and if those are equal, the lower odd card is compared.So for example 5-5-5-3-2 beats 4-4-4-K-5, which beats 4-4-4-Q-9, which beats 4-4-4-Q-8.

7. Two Pairs

6 7 8 9 10 Poker Hand

A pair consists of two cards of equal rank. In a hand with two pairs, the two pairs are of different ranks (otherwise you would have four of a kind), and there is an odd card to make the hand up to five cards. When comparing hands with two pairs, the hand with the highest pair wins, irrespective of the rank of the other cards - so J-J-2-2-4 beats 10-10-9-9-8 because the jacks beat the tens. If the higher pairs are equal, the lower pairs are compared, so that for example 8-8-6-6-3 beats 8-8-5-5-K. Finally, if both pairs are the same, the odd cards are compared, so Q-Q-5-5-8 beats Q-Q-5-5-4.

8. Pair

A hand with two cards of equal rank and three cards which are different from these and from each other. When comparing two such hands, the hand with the higher pair is better - so for example 6-6-4-3-2 beats 5-5-A-K-Q. If the pairs are equal, compare the highest ranking odd cards from each hand; if these are equal compare the second highest odd card, and if these are equal too compare the lowest odd cards. So J-J-A-9-3 beats J-J-A-8-7 because the 9 beats the 8.

9. Nothing

Five cards which do not form any of the combinations listed above. This combination is often called High Card and sometimes No Pair. The cards must all be of different ranks, not consecutive, and contain at least two different suits. When comparing two such hands, the one with the better highest card wins. If the highest cards are equal the second cards are compared; if they are equal too the third cards are compared, and so on. So A-J-9-5-3 beats A-10-9-6-4 because the jack beats the ten.

Hand Ranking in Low Poker

There are several poker variations in which the lowest hand wins: these are sometimes known as Lowball. There are also 'high-low' variants in which the pot is split between the highest and the lowest hand. A low hand with no combination is normally described by naming its highest card - for example 8-6-5-4-2 would be described as '8-down' or '8-low'.

It first sight it might be assumed that in low poker the hands rank in the reverse order to their ranking in normal (high) poker, but this is not quite the case. There are several different ways to rank low hands, depending on how aces are treated and whether straights and flushes are counted.

Ace to Five

This seems to be the most popular system. Straights and flushes do not count, and Aces are always low. The best hand is therefore 5-4-3-2-A, even if the cards are all in one suit. Then comes 6-4-3-2-A, 6-5-3-2-A, 6-5-4-2-A, 6-5-4-3-A, 6-5-4-3-2, 7-4-3-2-A and so on. Note that when comparing hands, the highest card is compared first, just as in standard poker. So for example 6-5-4-3-2 is better than 7-4-3-2-A because the 6 is lower than the 7. The best hand containing a pair is A-A-4-3-2. This version is sometimes called 'California Lowball'.

When this form of low poker is played as part of a high-low split variant, there is sometimes a condition that a hand must be 'eight or better' to qualify to win the low part of the pot. In this case a hand must consist of five unequal cards, all 8 or lower, to qualify for low. The worst such hand is 8-7-6-5-4.

Deuce to Seven

The hands rank in almost the same order as in standard poker, with straights and flushes counting and the lowest hand wins. The difference from normal poker is that Aces are always high , so that A-2-3-4-5 is not a straight, but ranks between K-Q-J-10-8 and A-6-4-3-2. The best hand in this form is 7-5-4-3-2 in mixed suits, hence the name 'deuce to seven'. The next best is 7-6-4-3-2, then 7-6-5-3-2, 7-6-5-4-2, 8-5-4-3-2, 8-6-4-3-2, 8-6-5-3-2, 8-6-5-4-2, 8-6-5-4-3, 8-7-4-3-2, etc. The highest card is always compared first, so for example 8-6-5-4-3 is better than 8-7-4-3-2 even though the latter contains a 2, because the 6 is lower than the 7. The best hand containing a pair is 2-2-5-4-3, but this would be beaten by A-K-Q-J-9 - the worst 'high card' hand. This version is sometimes called 'Kansas City Lowball'.

Ace to Six

Many home poker players play that straights and flushes count, but that aces can be counted as low. In this version 5-4-3-2-A is a bad hand because it is a straight, so the best low hand is 6-4-3-2-A. There are a couple of issues around the treatment of aces in this variant.

• First, what about A-K-Q-J-10? Since aces are low, this should not count as a straight. It is a king-down, and is lower and therefore better than K-Q-J-10-2.
• Second, a pair of aces is the lowest and therefore the best pair, beating a pair of twos.

It is likely that some players would disagree with both the above rulings, preferring to count A-K-Q-J-10 as a straight and in some cases considering A-A to be the highest pair rather than the lowest. It would be wise to check that you agree on these details before playing ace-to-six low poker with unfamiliar opponents.

Selecting from more than five cards

Note that in games where more than five cards are available, the player is free to select whichever cards make the lowest hand. For example a player in Seven Card Stud Hi-Lo 8 or Better whose cards are 10-8-6-6-3-2-A can omit the 10 and one of the 6's to create a qualifying hand for low.

Poker Hand Ranking with Wild Cards

A wild card card that can be used to substitute for a card that the holder needs to make up a hand. In some variants one or more jokers are added to the pack to act as wild cards. In others, one or more cards of the 52-card pack may be designated as wild - for example all the twos ('deuces wild') or the jacks of hearts and spades ('one-eyed jacks wild', since these are the only two jacks shown in profile in Anglo-American decks).

The most usual rule is that a wild card can be used either

1. to represent any card not already present in the hand, or
2. to make the special combination of 'five of a kind'.

This approach is not entirely consistent, since five of a kind - five cards of equal rank - must necessarily include one duplicate card, since there are only four suits. The only practical effect of the rule against duplicates is to prevent the formation of a 'double ace flush'. So for example in the hand A-9-8-5-joker, the joker counts as a K, not a second ace, and this hand is therefore beaten by A-K-10-4-3, the 10 beating the 9.

Five of a Kind

When playing with wild cards, five of a kind becomes the highest type of hand, beating a royal flush. Between fives of a kind, the higher beats the lower, five aces being highest of all.

The Bug

Some games, especially five card draw, are often played with a bug. This is a joker added to the pack which acts as a limited wild card. It can either be used as an ace, or to complete a straight or a flush. Thus the highest hand is five aces (A-A-A-A-joker), but other fives of a kind are impossible - for example 6-6-6-6-joker would count as four sixes with an ace kicker and a straight flush would beat this hand. Also a hand like 8-8-5-5-joker counts as two pairs with the joker representing an ace, not as a full house.

Wild Cards in Low Poker

In Low Poker, a wild card can be used to represent a card of a rank not already present in the player's hand. It is then sometimes known as a 'fitter'. For example 6-5-4-2-joker would count as a pair of sixes in normal poker with the joker wild, but in ace-to-five low poker the joker could be used as an ace, and in deuce-to-seven low poker it could be used as a seven to complete a low hand.

Lowest Card Wild

Some home poker variants are played with the player's lowest card (or lowest concealed card) wild. In this case the rule applies to the lowest ranked card held at the time of the showdown, using the normal order ace (high) to two (low). Aces cannot be counted as low to make them wild.

Double Ace Flush

Some people play with the house rule that a wild card can represent any card, including a duplicate of a card already held. It then becomes possible to have a flush containing two or more aces. Flushes with more than one ace are not allowed unless specifically agreed as a house rule.

Natural versus Wild

Some play with the house rule that a natural hand beats an equal hand in which one or more of the cards are represented by wild cards. This can be extended to specify that a hand with more wild cards beats an otherwise equal hand with fewer wild cards. This must be agreed in advance: in the absence of any agreement, wild cards are as good as the natural cards they represent.

Incomplete Hands

In some poker variants, such as No Peek, it is necessary to compare hands that have fewer than five cards. With fewer than five cards, you cannot have a straight, flush or full house. You can make a four of a kind or two pairs with only four cards, triplets with three cards, a pair with two cards and a 'high card' hand with just one card.

The process of comparing first the combination and then the kickers in descending order is the same as when comparing five-card hands. In hands with unequal numbers of cards any kicker that is present in the hand beats a missing kicker. So for example 8-8-K beats 8-8-6-2 because the king beats the 6, but 8-8-6-2 beats 8-8-6 because a 2 is better than a missing fourth card. Similarly a 10 by itself beats 9-5, which beats 9-3-2, which beats 9-3, which beats a 9 by itself.

Ranking of suits

In standard poker there is no ranking of suits for the purpose of comparing hands. If two hands are identical apart from the suits of the cards then they count as equal. In standard poker, if there are two highest equal hands in a showdown, the pot is split between them. Standard poker rules do, however, specify a hierarchy of suits: spades (highest), hearts, diamonds, clubs (lowest) (as in Contract Bridge), which is used to break ties for special purposes such as:

• drawing cards to allocate players to seats or tables;
• deciding who bets first in stud poker according to the highest or lowest upcard;
• allocating a chip that is left over when a pot cannot be shared exactly between two or more players.

I have, however, heard from several home poker players who play by house rules that use this same ranking of suits to break ties between otherwise equal hands. For some reason, players most often think of this as a way to break ties between royal flushes, which would be most relevant in a game with many wild cards, where such hands might become commonplace. However, if you want to introduce a suit ranking it is important also to agree how it will apply to other, lower types of hand. If one player A has 8-8-J-9-3 and player B has 8-8-J-9-3, who will win? Does player A win by having the highest card within the pair of eights, or does player B win because her highest single card, the jack, is in a higher suit? What about K-Q-7-6-2 against K-Q-7-6-2 ? So far as I know there is no universally accepted answer to these questions: this is non-standard poker, and your house rules are whatever you agree that they are. Three different rules that I have come across, when hands are equal apart from suit are:

1. Compare the suit of the highest card in the hand.
2. Compare the suit of the highest paired card - for example if two people have J-J-7-7-K the highest jack wins.
3. Compare the suit of the highest unpaired card - for example if two people have K-K-7-5-4 compare the 7's.

Although the order spades, hearts, diamonds, clubs may seem natural to Bridge players and English speakers, other suit orders are common, especially in some European countries. Up to now, I have come across:

• spades (high), hearts, clubs, diamonds (low)
• spades (high), diamonds, clubs, hearts (low)
• hearts (high), spades, diamonds, clubs (low) (in Greece and in Turkey)
• hearts (high), diamonds, spades, clubs (low) (in Austria and in Sweden)
• hearts (high), diamonds, clubs, spades (low) (in Italy)
• diamonds (high), spades, hearts, clubs (low) (in Brazil)
• diamonds (high), hearts, spades, clubs (low) (in Brazil)
• clubs (high), spades, hearts, diamonds (low) (in Germany)

As with all house rules, it would be wise to make sure you have a common understanding before starting to play, especially when the group contains people with whom you have not played before.

Stripped Decks

In some places, especially in continental Europe, poker is sometimes played with a deck of less than 52 cards, the low cards being omitted. Italian Poker is an example. As the pack is reduced, a Flush becomes more difficult to make, and for this reason a Flush is sometimes ranked above a Full House in such games. In a stripped deck game, the ace is considered to be adjacent to the lowest card present in the deck, so for example when using a 36-card deck with 6's low, A-6-7-8-9 is a low straight.

Playing poker with fewer than 52 cards is not a new idea. In the first half of the 19th century, the earliest form of poker was played with just 20 cards - the ace, king, queen, jack and ten of each suit - with five cards dealt to each of four players. The only hand types recognised were, in descending order, four of a kind, full house, three of a kind, two pairs, one pair, no pair.

No Unbeatable Hand

In standard poker a Royal Flush (A-K-Q-J-10 of one suit) cannot be beaten. Even if you introduce suit ranking, the Royal Flush in the highest suit is unbeatable. In some regions, it is considered unsatisfactory to have any hand that is guaranteed to be unbeaten - there should always be a risk. There are several solutions to this.

In Italy this is achieved by the rule 'La minima batte la massima, la massima batte la media e la media batte la minima' ('the minimum beats the maximum, the maximum beats the medium and the medium beats the minimum'). A minimum straight flush is the lowest that can be made with the deck in use. Normally they play with a stripped deck so for example with 40 cards the minimum straight flush would be A-5-6-7-8 of a suit. A maximum straight flush is 10-J-Q-K-A of a suit. All other straight flushes are medium. If two players have medium straight flushes then the one with higher ranked cards wins as usual. Also as usual a maximum straight flush beats a medium one, and a medium straight flush beats a minimum one. But if a minimum straight flush comes up against a maximum straight flush, the minimum beats the maximum. In the very rare case where three players hold a straight flush, one minimum, one medium and one maximum, the pot is split between them. See for example Italian Poker.

In Greece, where hearts is the highest suit, A-K-Q-J-10 is called an Imperial Flush, and it is beaten only by four of a kind of the lowest rank in the deck - for example 6-6-6-6 if playing with 36 cards. Again, in very rare cases there could also be a hand in the showdown that beats the four of a kind but is lower than the Imperial Flush, in which case the pot would be split.

Hand probabilities and multiple decks

The ranking order of poker hands corresponds to their probability of occurring in straight poker, where five cards are dealt from a 52-card deck, with no wild cards and no opportunity to use extra cards to improve a hand. The rarer a hand the higher it ranks.

This is neither an essential nor an original feature of poker, and it ceases to be true when wild cards are introduced. In fact, with a large number of wild cards, it is almost inevitable that the higher hand types will be the commoner, not rarer, since wild cards will be used to help make the most valuable type of hand from the available cards.

Mark Brader has provided probability tables showing the frequency of each poker hand type when five cards are dealt from a 52-card deck, and also showing how these probabilities would change if multiple decks were used.