Poker Hands - Rank and Frequency

Most people know right away that a Royal Flush is the highest poker hand, and that Straight Flushes are next. Sometimes people confuse the order of the hands that come next, particularly the ones that are rather rare. This is a refresher:

Rank of Hands

Four of a Kind (Quads). Beats everything except a Royal or a Straight Flush. In a seven-card hand, suppose you come up with KKKK777. The five best cards are a quad of kings and a 7 kicker. The fact that you can see three of them is not relevant.

Full House (Boat). Trips and a pair. The expression is "[trips] full of [the pair]" as in "Aces full of Tens." (A-A-A-10-10). Ties are resolved in favor of the rank of the trips.

Flush. Five cards in the same suit. The tiebreaker in flushes is the rank of the highest card in the flush. If that is also a tie, then by the second-highest card, and so on. Suit hierarchy is irrelevant.

Straight. Five sequential ranks, regardless of suit. Aces can be high or low. A low-ace straight is called a "wheel" or a "bicycle." Ties in straights are resolved according to the highest card of the sequence.

Three of a kind (Trip or Set). Three cards of the same rank. (The term "set" is reserved for trips when one or two of the cards is face down.)

Two Pair. Two groups of cards, in which each group has two cards of the same rank. Ties are resolved according to the higher of the two pair, and if that is tied, by the lower, and if that is tied, by the rank of the kicker.

One Pair. A group of two cards of the same rank. Ties are resolved by the rank of the higher pair, then by the highest single card in the hand. If that is tied, then by the second-highest singleton in the hand, and so on.

High Card. The highest singleton in the (otherwise) scoreless hand. Ties are resolved as for a pair.

Suit Rank. Suit Rank is relevant only in the context of determining who the "bring-in" player is at the beginning of a game of Seven-Card Stud. The player with the lowest ranked door card is that person. Ties in rank are resolved by suit, in descending order: Spades, Hearts, Diamonds, Clubs. Otherwise, suits are irrelevant to hierarchy, and are of concern only in flushes.

Frequency of Hands

It might be useful to know what the chances are that a given hand might be dealt out. To look at the odds in this way requires suspending reality, as real odds depend on what cards have already been played, how many players are still in, and what cards are visible. In real life, a player might fold a hand before getting to see what it would have been. In a real game, for example, the odds that a given card will be dealt to you will vary according to whether the card (or its rank or suit-mate) is visible in some other player's hand. The following figures simply report the frequency of hands expected when one turns up seven cards randomly out of a 52-card shuffled deck.

There are 133,784,560 different combinations of seven cards in a 52-card deck. That is the denominator (for seven-card games, obviously).

For a Royal Flush, there are 4 possible outcomes, one for each suit, and they occupy 5 of the seven cards. The other cards can be anything. Thus, there are 4140 different sets of seven cards that can contain one of the four royal flushes. This yields a chance of one in every 32,175 deals.

For a Straight Flush of any kind (including a Royal), there are ten times as many possibilities, so the chances improve to about once in every 3,217 deals.

More realistically, Four of a kind will come along about once in every 595 deals. A full house can be expected about once in every 39 deals, and a flush about once in 33 deals. Straights would show up about once in every 22 deals, three of a kind every 21 deals, two pair about once in four deals, a pair about once in 2.25 deals, and a high card only about once in six hands.

Drawing Odds

Suppose instead that you have a certain hand and want to know what the odds are that the River will make the hand. Also assume that the visible cards do not tell you that the card(s) you are hoping for are "dead" (already played).

The odds of "making" a flush on the river with four to a flush in hand are 1 in 4.5.

The odds of "making" a double-ended open straight are 1 in 5.

The offs of "making" an inside straight or single-ended open straight are 1 in 11.

The double open-ended straight flush is a long shot: 1 in 23, and 1 in 46 for the inside straight flush.

If you have three of a kind, and are hoping for a full house on the River, the chances are 2:1 against a second pair (40%).