Poker Basics
Purpose
This document is meant as a brief, but somewhat detailed, introduction to (or refresher on) the game of poker. It covers the terminology I use in lectures, as well as the mechanics of a poker deck.
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Poker Deck
A standard poker deck has \( 52 \) cards. Each card has a suit and a value. We often abbreviate a card from the deck as \( vs \), where \( v \) is the symbol for its value, and \( s \) is the symbol for its suit. When read aloud, this is "the \( v \) of \( s \)".
Suits
The suits are given in the table below.
| Name | Symbol | Picture |
|---|---|---|
| Clubs | \( C \) | \( \clubsuit \) |
| Hearts | \( H \) | \( \heartsuit \) |
| Spades | \( S \) | \( \spadesuit \) |
| Diamonds | \( D \) | \( \diamondsuit \) |
Values
The values, listed from highest to lowest worth, are given in the table below.
| Name | Symbol | Numeric |
|---|---|---|
| Ace | \( A \) | 1 |
| King | \( K \) | 13 |
| Queen | \( Q \) | 12 |
| Jack | \( J \) | 11 |
| Ten | \( T \) | 10 |
| Nine | \( 9 \) | 9 |
| Eight | \( 8 \) | 8 |
| Seven | \( 7 \) | 7 |
| Six | \( 6 \) | 6 |
| Five | \( 5 \) | 5 |
| Four | \( 4 \) | 4 |
| Three | \( 3 \) | 3 |
| Two | \( 2 \) | 2 |
Notice the distinction between the worth of a value and it's numeric equivalent. The Ace is the highest value, but has the lowest numeric equivalent. This will be important for ranking poker hands later.
Hands
We will analyze five card poker. A hand is a set of five cards. Note that the hand does not depend on order! Each hand has a rank. Below are all the possible hand ranks, in order from highest worth to least worth.
If a hand fits multiple descriptions, it's rank is the one with the highest worth.
The hand \( \{ 5S, 5H, 2C, 2S, 2D \} \) is a full house. While it also satisfies the conditions to be a Pair in value \( 5 \), that is a lower-rank hand.
Note that these hand ranks are comparable within themselves, e.g., some straights are worth more than other straights. This is done by comparing the cards satisfying (corresponding parts of) the description first, and then comparing the remaining cards in order. The higher value always wins.
The hands \( G_1 = \{ KS, QD, TC, TD, 4C \} \), \( G_2 = \{ KC, TH, TD, 9H, 8C \} \), and \( G_3 = \{ AC, AD, 5C, 4H, 3D \} \) are all pairs, but they are ranked \( G_3 > G_1 > G_2 \). Note \( G_3 > G_1 \) because when comparing the pairs, their values are \( A > T \).
To compare \( H_1 \) with \( H_2 \), we proceed as follows.
- The pairs are in value \( T = T \), so look at the other cards.
- The greatest non-pair cards are of value \( K = K \), so keep looking.
- The next-greatest non-pair cards are of value \( Q > 9 \), so we stop here declaring \( G_1 > G_2 \).
Royal Flush
A straight flush with an Ace and a Ten.
For example, \( G = \{ AD, KD, QD, JD, TD \} \) is a Royal Flush (in diamonds).
Straight Flush
Four-of-a-Kind
Four cards all of the same value.
For example, \( G = \{ QD, 5C, 5H, 5S, 5D \} \) is a four-of-a-kind (of fives).
Full House
A pair and a three-of-a-kind with different values.
For example, \( G = \{ JS, JD, 2C, 2H, 2D \} \) is a full house (twos over jacks).
Flush
Five cards all of the same suit.
For example, \( G = \{ KH, JH, 7H, 6H, 2H \} \) is a flush (in hearts).
Royal Straight
A straight with an Ace and a Ten.
For example, \( G = \{ AH, KS, QD, JD, TH \} \) is a royal straight.
Straight
Five cards with consecutive values. For these purposes, the Ace can either be one greater than a King or one less than a Two, but it cannot be both.
For example, \( G_1 = \{ 9C, 8H, 7H, 6S, 5H \} \) is a straight (running five-to-nine) and \( G_2 = \{ 9C, 8H, 7H, 6S, 5H \} \) is a straight (running ace-to-five). Note that\( G_3 = \{ 2C, AH, KH, QS, JH \} \) is not a straight by these rules—it is a high card (ace) hand.
Three-of-a-kind
Three cards with the same value.
For example, \( G = \{ 7S, 6C, 6H, 6S, 4D \} \) is a three-of-a-kind (of sixes).
Two Pair
Two pairs of cards with (independently) the same ranks.
For example, \( G = \{ KD, 8C, 8S, 3S, 3D \} \) is a two-pair (of fours over twos).
Pair
Two cards with a common rank.
For example, \( G = \{ AS, KD, JH, 2C, 2D \} \) is a pair (of twos).
High Card
Any hand which does not fit any of the other criteria above.
For example, \( G = \{ 7S, 5S, 4C, 3H, 2D \} \) is a high card (seven) hand.