My Math Forum Grid Based Poker odds question

 February 12th, 2012, 11:24 AM #1 Newbie   Joined: Feb 2012 Posts: 1 Thanks: 0 Grid Based Poker odds question I'm developing a game that uses a grid based poker system and I need to figure out the odds of drawing a hand in order to balance the game properly. The game starts with 54 cards ( this includes jokers ) To start the game, you deal 9 cards in the center of a 5x5 grid. ( X = Grid Space ) ( Y = Dealt Card ) x x x x x x Y Y Y x x Y Y Y x x Y Y Y x x x x x x There are two players. Each player now draws two cards from the deck. 54 - 9 - 2 -2 = 41 cards left in the deck. The starting player draws two more cards bringing the deck total down to 39 cards. The starting player now uses two cards from their hand of four to place across from each other horizontally, vertically or diagonally to create a 5 card hand of poker. The corner cards must be placed across from each other on a diagonal. Example : x x x x x Y Y Y Y Y <---- played hand x Y Y Y x x Y Y Y x x x x x x x x x Y x x Y Y Y x x Y Y Y x x Y Y Y x x x x Y x ^ played hand Y x x x x x Y Y Y x x Y Y Y x x Y Y Y x x x x x Y <-- Played hand \ Using this system, does anyone know what or how to figure out the odds for drawing each type of poker hand ( pair, two pair, three of a kind, 4 of a kind, Full House, Straight, Flush, Straight Flush, Royal Flush, 5 of a kind ( using wild cad ) Any data would be appreciated !!!
February 12th, 2012, 03:38 PM   #2
Math Team

Joined: Dec 2006
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Re: Grid Based Poker odds question

Hello, convicsik!

Quote:
 I'm developing a game that uses a grid based poker system and I need to figure out the odds of drawing a hand in order to balance the game properly. The game starts with 54 cards ( this includes jokers). To start the game, you deal 9 cards in the center of a 5x5 grid. ($x$ = Grid Space) ($Y$ = Dealt Card) [color=beige]. . [/color]$\begin{array}{ccccc} x&x&x&x&x \\ \\ x&Y&Y&Y&x \\ \\ x&Y&Y&Y&x \\ \\ x&Y&Y&Y&x \\ x&x&x&x&x \end{array}$ There are two players. Each player now draws two cards from the deck. [color=beige]. . [/color]54 - 9 - 2 -2 = 41 cards left in the deck. The starting player draws two more cards bringing the deck total down to 39 cards. The starting player now uses two cards from their hand of four to place across from each other horizontally, vertically or diagonally to create a 5 card hand of poker. The corner cards must be placed across from each other on a diagonal. Examples: [color=beige]. . [/color]$\begin{array}{cccccccc} x&x&x&x&x \\ \\ Y&Y&Y&Y&Y & \leftarrow & \text{hand} \\ \\ x&Y&Y&Y&x \\ \\ x&Y&Y&Y&x \\ \\ x&x&x&x&x \end{array}$ [color=beige]. . [/color]$\begin{array}{ccccc} x&x&x&Y&x \\ \\ x&Y&Y&Y&x \\ \\ x&Y&Y&Y&x \\ \\ x&Y&Y&Y&x \\ \\ x&x&x&Y&x \\ \\ \\ &&&\uparrow \end{array}$ [color=beige]. . . . . . . [/color]$\text{hand}$ [color=beige]. . [/color]$\begin{array}{cccccc} Y&x&x&x&x \\ \\ x&Y&Y&Y&x \\ \\ x&Y&Y&Y&x \\ \\ x&Y&Y&Y&x \\ \\ x&x&x&x&Y \\ \\ \\ &&&&& \nwarrow \text{ hand}\end{array}$ Using this system, does anyone know what or how to figure out the odds for drawing each type of poker hand?[color=beige] .[/color](One Pair, Two Pair, Three of a Kind, Straight, Flush, Full House, Straight Flush, Royal Flush, 5 of a Kind (using wild cards).

The problem may be too immense for us carbon-based lifeforms.

With any 3-by-3 array, we have 8 possible three-card hands to examine and try to improve.

I have done an analysis on three-card hands.

[color=beige]. . [/color]$\begin{array}{cc}\text{Str. flush} & \;\;\;\;48 \\ \\ \text{3-of-a-kind} & \;\;\;\;52 \\ \\ \text{Straight} & \;\;720 \\ \\ \text{Flush} & \;1,144 \\ \\ \text{One pair} & \;3,744 \\ \text{Washout} & 16,392 \\ \\ \hline \\ \text{Total:} & 22,100 \end{array}$

I'll let you work out the rest . . .

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