## Texas Hold Dem Poker Cards, Game, and Probability, numerical analyis

This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Its very hard to reply reasonably without some background of the correspondent on their Wiki bio page. Thanks,gold

### Preface

gold Here are some calculations for poker probability using TCL expressions in the Windows console with the easy eye line calculator. The impetus for these calculations was checking some poker probabilities, mostly on Texas Holdem Poker, but also 5 Card Poker. Its easy enough to cut and paste TCL based formulas into the easy eye calculator console. Most of the checked testcases involve TCLLIB math functions.

### Texas Holdem Poker into TCL.

The baby game engine in Mahjong Deletion can be converted or hacked into a set of 52 poker cards. In Texas Holdem Poker, there are four stages of deals and four stages of betting until the final round. All suits are equal and no suit is worth more than another. No Joker cards are used. The winner has the best five-card hand possible using the player’s two cards as pocket cards and the five dealt in the middle of the table. The odds change with each deal, so betting is tricky for the players or exciting for the spectators. On most deals, players have betting options for check, call, raise, or fold. The end of game or last betting session is sometimes called a showdown. Usually a chip marked dealer is passed or shared among the players for the responsibility of dealing the cards. Two antes or blinds are placed for the responsibility of different players as the places of the binds are relative to the selected dealer for the hand. The "Big Blind" is placed on the right of the dealer and the "Small Blind" is placed on the left of the dealer. For only two players or small player games, some of the "Big Blind" and "Small Blind" rules are omitted or handled differently. On the canvas widget of TCL, the four stages of deals can be raised or lowered into spectator view from the underscreen with buttons and keyed canvas tags. In the internal TCL code, each card can be dealt from a fresh deck of electrons. Initially and unlike the Las Vegas tables, a dealt card will not be removed from the 52 total deck.

### Testcases are important in development of TCL programs.

Revision

In planning any software, it is advisable to gather a number of testcases to check the results of the program. The math for the testcases can be checked by pasting statements with added brackets in the TCL console. Also, some of the pseudocode statements were checked in the google search engine which will take math expressions. Ref factorial procedure in original routines was posted by Suchenwirth RS, but the factorial and derived choose functions are included in the TCLLIB math library.

### Texas Holdem Poker

Revision. In Texas Holdem Poker, there are four stages of deals and four stages of betting until the final round. All suits are equal and no suit is worth more than another. No Joker cards are used. Gameplay The winner has the best five-card hand possible using the player's two cards as pocket cards and the five dealt in the middle of the table. The odds change with each deal, so betting is tricky for the players or exciting for the spectators. On most deals, players have betting options for check, call, raise, or fold. The end of game or last betting session is sometimes called a showdown. Betting and Blinds Usually a chip marked dealer is passed or shared among the players for the responsibility of dealing the cards. Two antes or blinds are placed for the responsibility of different players as the places of the binds are relative to the selected dealer for the hand. The "Big Blind" is placed on the right of the dealer and the "Small Blind" is placed on the left of the dealer. For only two players or small player games, some of the "Big Blind" and "Small Blind" rules are omitted or handled differently. Implementation in Mahjong Deletion On the canvas widget of TCL, the four stages of deals can be raised or lowered into spectator view from the underscreen with buttons and keyed canvas tags. In the internal TCL code, each card can be dealt from a fresh deck of electrons. Initially and unlike the Las Vegas tables, a dealt card will not be removed from the 52 total deck.

### Table 1: Betting Gist

table. Four styles of Texas Holdem odds dealt cards in a hand Bet numbers player options comment if any
preflop odds 2 bet1 After pocket, call, raise, or fold first 2 called pocket or hole card
flop odds 3 bet2 After flop, check, raise, or fold
turn odds 1 bet3 After turn, check, raise, or fold
river odds 1 bet4 After river, check, raise, or fold
>> gist <<<
totals 7 cards 4 bets 12 possible actions, expected in typical game

### Table 2: Probabilities in Texas Holdem

quantity P_probability, S_space etc 100X, alternate percent, if applicable formula TCL comments if any
texas_pocket_pair_space 1326 expr { math::choose 52 2 }
texas_pocket_pair 0.004524886 0.4524886 expr { (1.*math::choose 4 2 )/ math::choose 52 2 }odds 220:1
any_texas_pocket_pair 0.0588 5.88 expr { 78/1326 } odds 16:1
specific_texas_suited combination 4 expr { math::choose 4 1 }
specific_texas_suited_hand 0.003 0.3 expr { 4/1326 } odds 331:1
any_texas_suited combination 312 expr { 4*78 }
any_texas_suited_hand 0.2352 23.52 expr { 312/1326 } odds 3.25:1

### Table 3: Probabilities in 5 Card Poker

Picking up a five card hand, what is the probability and odds of a straight flush, less royal flush ? The sample space is 52!/((5!* (52-5)!) or 2598960. The frequency is (10 choose 1)*(4 choose 1) -(4 choose 1) , 10!/((1!* (10-1)!) * 4!/((1!* (4-1)!) - 4!/((1!* (4-1)!) , 40-4,or 36. The text books use the expression <4 choose 1>, but the TCLLIB math library is invoked by the expression <math::choose 4 1>.

```set straight_flush  [expr { (1.*36)/ 2598960    } ]
set straight_flush_odds  [expr { 1./( (1.*36)/ 2598960 )   } ]
answer%  1.39E-5 ,  72192.0```
Probabilities in 5 Card Poker
hand choose formula frequency division terms probability odds
royal flush (4 choose 1) 4 4/258960 1.54E-5 64739
straight flush less royal flush (10 choose 1)* (4 choose 1) -(4 choose 1) 36 36/258960 1.39E-572192
four of kind (13 choose 1)* (12 choose 1) *(4 choose 1) 624 624/258960 2.4E-44164
full house (13 choose 1)* (4 choose 3) *(12 choose 2) 858 858/258960 2.11E-246.3
three of kind (13 choose 1)* (4 choose 3) *(12 choose 2)* (4 choose 2) 156 156/258960 1.44E-3693

### References:

• google < Texas Poker Wikipedia >
• Wikipedia search engine < Texas Poker algorithm >
• Wikipedia search engine < multiplication algorithm >
• Wikipedia search engine < Texas Poker Math >
• Wikipedia search engine < Texas Poker lograrithms >
• Wikipedia search engine <Texas Poker >
• Wolfram Alpha search engine < Texas Poker >
• Mahjong_Style_Deletion
• Poker Probability and Calculator Demo Example
• precision math procedure used in TCL calculator Arjen Markus AM
• Computers and real numbers
• Floating-point formatting
• Texas Hold ’em poker solved by computer Science, AAAS.
• Sklansky, David (2005). The Theory of Poker (Fourth ed.). Las Vegas: Two plus two.
• Harrington, Dan and Bill Robertie (2004). Harrington on Hold'em:
• Expert Strategy For No-Limit T ournaments; Volume I: Strategic Play. Two Plus Two Publications.
• D. Purdy, The Illustrated Guide to Texas Hold’em, Sourcebooks, Inc., 2005.
• pokerproj
• Wikipedia, Poker_probability.pdf
• Wikipedia, Texas_hold_'em.pdf
• Poker probability (Texas hold 'em)
• Estimating Winning Probability for Texas Hold'em Poker,
• Wenkai Li and Lin Shang
• Probabilities in Texas Hold'em,
• The mathematics of Texas Hold'em Poker Probability-based analyses on concrete hands.
• Alspach's Mathematics and Poker Page
• Probabilities in Texas Hold'em, Brian Alspach, 22 August 2001
• Probabilities in Texas Hold'em
• Hold'Em Solitaire , on wiki, Keith Vetter 2005-10-26

### Pseudocode & Equations Section

```        more than one formula for 1) tables and 2) calculator shell
```

## Appendix Code

### Trial Shell

```        ######))))))  TCL source code follows
# pretty print from autoindent and ased editor
# program Texas Hold Dem Cards.
# adapted from Mahjong_Style_Deletion
# written on Windows XP for TCL
# working under TCL version 8.6
# The original base M. was adapted
# from aricb on the Ask page.
# Probably could start a post,
# since it is a good baby game
# engine. Wish to study dealt values and
# probability of Texas Hold Dem Cards.
# Need to evaluate hands and bet on them.
# I added a few buttons at the bottom
# and tried to keep the clean code separate up top.
# Richard Suchenwirth RS wrote lpick routine below.
# de on  TCL Club, 15nov2018
package require Tk
package require math
namespace path {::tcl::mathop ::tcl::mathfunc }
global deck suit
global selectedid selectedtext
global new_card_state
set new_card_state 0
set deck {}
set suitfrenchblack { \u2665 \u2666 \u2663 \u2660 }
set suitfrenchblanc { \u2664 \u2661 \u2662 \u2667 }
# utilities
# suchenwirth_subroutine RS
proc lpick L {lindex \$L [expr {int(rand()*[llength \$L])} ];}
proc plainsub {text item replacewith} {
set text [string map [list \$item \$replacewith] \$text]
}
proc deal {args} {
global deck suit
foreach suit { \u2665 \u2666 \u2663 \u2660  } {
foreach item \$args {
set kay "\$item \$suit "
set value 1
if {\$item == "J"} {set value 10}
if {\$item == "Q"} {set value 10}
if {\$item == "K"} {set value 10}
if {\$item == "A"} {set value 10}
set kay "[append item \$suit \ \$value ]"
lappend deck \$kay
}}
return  \$deck
}
deal 2 3 4 5 6 7 8 9 10 A J Q K
puts "\$deck new_card state equals \$new_card_state"
proc self_help {} {
set msg "Texas Hold Dem Cards V2
from TCL ,
# self help listing
# >>>>  Copyright Notice  <<<<<
# This posting, screenshots, and TCL source code is
# copyrighted under the TCL/TK 8.6 license terms.
# Editorial rights retained under
# the TCL/TK license terms
# and will be defended as necessary in court.
# >>>>   Push Button Operation <<<<
# Push left mouse to activate buttons.
# Deal shuffles all hands.
# Move_right_mouse_for_one_card deal.
# All cards listed with value.
# Deck is 52 cards, no jokers.
# Report allows copy and paste
# from console to conventional
# texteditor. For testcases,
# the testcase number is internal
# to the calculator and
# will not be printed until
# the report button is pushed
# for the current result numbers.
# Where the program defaults are recalled,
# canvas colors/features/variables/globals/calculations/clear displays
# may reset prior to calculation
# ( in push button list) .
# Buttons on the calculator are not
# limited to a single action procedure
# or single shot. A button command may
# call multiple statements and procs,
# effectively a separate program or
# subprogram installed as
# a list of procedures in a button.
# Alternately, the list of procedures
# need not be installed in a button,
# but in a separate procedure.
Conventional text editor formulas
or  grabbed from internet
screens can be pasted into green console.
Try copy and paste following
into green screen console
set answer [* 1. 2. 3. 4. 5. ]
returns  120
# gold on  TCL Club, 15nov2018 "
tk_messageBox -title "self_help" -message \$msg }
proc populateCanvas {canvas cols rows} {
variable ids ;# links text ids with respective rect ids
variable boxes ;# lists text id and text associated with each rect id
global deck suitfrenchblack suitfrenchblanc
catch {unset ids boxes}
# parameters for drawing boxes
set boxwidth 50
set boxheight 20
set padx 10
set pady 10
set colors {red orange yellow green blue purple}
set money_color { red  }
set labels {one two three four}
# draw the boxes
for {set row 0} {\$row < \$rows} {incr row} {
for {set col 0} {\$col < \$cols} {incr col} {
# calculate coordinates
set x1 [expr {\$col * (\$boxwidth + \$padx) + \$padx}]
set x2 [expr {\$x1 + \$boxwidth}]
set x3 [expr {\$x1 + (\$boxwidth / 2)}]
set y1 [expr {\$row * (\$boxheight + \$pady) + \$pady}]
set y2 [expr {\$y1 + \$boxheight}]
set y3 [expr {\$y1 + (\$boxheight / 2)}]
# choose color and text
set color [lindex \$colors [expr {int(rand() * [llength \$colors])}]]
#set text [lindex \$labels [expr {int(rand() * [llength \$labels])}]]
set text [lindex \$deck [expr {int(rand() * [llength \$deck])}]]
# create the boxes
set boxid [\$canvas create rectangle \$x1 \$y1 \$x2 \$y2 -fill \$color -outline black]
set textid [\$canvas create text \$x3 \$y3 -text \$text]
# remember which text item goes with which box and what the text says
set boxes(\$boxid) [list \$textid \$text]
set ids(\$textid) \$boxid
set ids(\$boxid) \$boxid
}
}
}
proc clearSelection {canvas} {
# if there is an existing currently selected box, change its border back to
# black; set variables selectedid and selectedtext to empty strings
variable selectedid
variable selectedtext
if {\$selectedid ne "" && [\$canvas find withtag \$selectedid] ne ""} {
\$canvas itemconfigure \$selectedid -outline black
}
set selectedid ""
set selectedtext ""
}
proc handleSelection {canvas id} {
# determine the "next step" by comparing the newly selected box id and text
# to the box id and text of the previously selected box (contained in
# the variables \$selectedid and \$selectedtext)
variable ids
variable boxes
variable selectedid
variable selectedtext
set boxid \$ids(\$id)
lassign \$boxes(\$boxid) textid text
if {\$boxid eq \$selectedid} {
# user has selected the same box twice; deselect the box
clearSelection \$canvas
return
} elseif {\$text eq \$selectedtext} {
# user has selected two boxes with the same text; delete both boxes
\$canvas delete \$boxid \$textid \$selectedid [lindex \$boxes(\$selectedid) 0]
clearSelection \$canvas
return
} else {
# user has selected a box when there is no selection, or user has
# selected a box with different text than the previously selected box;
# try to deselect the previous box, then select the new box
clearSelection \$canvas
set selectedid \$boxid
set selectedtext \$text
\$canvas itemconfigure \$boxid -outline white
return
}
}
proc processClick {canvas x y} {
# determine whether the user has
# clicked on something; if so, call
# proc handleSelection
set id [\$canvas find overlapping \$x \$y \$x \$y]
if {[llength \$id] > 0} {
handleSelection \$canvas [lindex \$id 0]
}
}
set win [toplevel .demo]
wm title \$win "Texas Hold Dem Cards"
set canvas [canvas \$win.canvas]
pack \$canvas  -expand 1 -fill both
bind \$canvas <Button-1> [list processClick %W %x %y]
set state 1
proc deal2 {} {
global selectedid selectedtext
global coloritem canvas details
set coloritem gray
# mahjongstyle { \$state }
populateCanvas \$canvas 7 5
set selectedid ""
set selectedtext ""
}
proc uncover {} {
global coloritem canvas details
global deal1 deal2 deal3 deal4 deal5 deal6 deal7
set state2 1
#\$canvas create rectangle 10 10 500 300 -fill blue -outline red -tag deal1
if { \$state2 == 5 } { \$canvas raise deal1 ;}
if { \$state2 == 1 } { \$canvas lower deal1 ;}
}
proc flip_new_card {} {
global deck suit canvas  x y
global selectedid selectedtext
global new_card_state
# determine whether the user has clicked on something;
# if so, flip to random new card
# set id [\$canvas find overlapping \$x \$y \$x \$y]
set id ""
if {[llength \$id] > 0} {
configure \$id -text "switch"
}
}
proc mahjongstyle { state } {
global coloritem canvas details
set coloritem gray
set details {
# program Texas Hold Dem Cards.
# adapted from Mahjong_Style_Deletion
# written on Windows XP for TCL
# working under TCL version 8.6
# The original base M. was adapted
# from aricb on the Ask page.
# Probably could start a post,
# since it is a good baby game
# engine. Wish to study value and
# probability of Texas Hold Dem cards.
# Need to evaluate hands and bet on them.
# I added a few buttons at the bottom
# and tried to keep the clean code separate up top.
Mahjong-style deletion.
press in succession two tiles with the same text.
both tiles should be deleted.
The Mahjong-style deletion is for
the right mouse press.}
\$canvas create window 100 200 -window [button \$canvas.a -width 10 -text deal \
-bg \$coloritem    -command {deal2;}]
\$canvas create window 100 250 -window [button \$canvas.b -width 10 -text swap_card \
-bg \$coloritem    -command {swap_new_cards 2 ;}]
\$canvas create window 200 200 -window [button \$canvas.c -width 10 -text about \
-bg \$coloritem    -command {tk_messageBox -message \$details}]
\$canvas create window 200 250 -window [button \$canvas.d -width 10 -height 1 -text report \
-bg \$coloritem     -command {puts "new_card_state equals \$new_card_state"; console show }]
\$canvas create window 300 200 -window [button \$canvas.e -width 10 -text self_help \
-bg \$coloritem    -command {self_help}]
\$canvas create window 300 250 -window [button \$canvas.f -width 10 -height 1 -text exit \
-bg \$coloritem     -command {exit }]
\$canvas create window 400 200 -window [button \$canvas.g -width 10 -text uncover_3-5 \
-bg \$coloritem    -command {uncover}]
\$canvas create window 400 250 -window [button \$canvas.h -width 10 -height 1 -text uncover_6-7 \
-bg \$coloritem     -command {uncover }]
}
mahjongstyle { \$state }
populateCanvas \$canvas 7 5
set selectedid ""
set selectedtext ""
\$canvas configure -background aquamarine4 -highlightcolor brown -relief raised -border 10
proc swap_new_cards { state } {
global new_card_state
set new_card_state 1
flip_new_card
return 1
}
proc moveobject {object x y} {
global deck suitfrenchblack suitfrenchblanc
global canvas
\$canvas coords \$object [expr \$x-25] [expr \$y-25]
set new_card  [ lpick \$deck ]
puts \$new_card
\$canvas itemconfigure \$object -text \$new_card -fill orange
}
# set boxid [\$canvas create rectangle \$x1 \$y1 \$x2 \$y2 -fill \$color -outline black]
set egg [\$canvas create text 550  200 -text move_right_mouse_for_new_card -fill red ]
\$canvas bind \$egg <B3-Motion>  {moveobject \$egg %x %y}
console hide
console eval {wm title . "Texas Hold Dem Cards, screen grab and paste from console 2 to texteditor"}
console eval {.console config -bg palegreen}
console eval {.console config -font {fixed 20 bold}}
console eval {wm geometry . 40x20}
console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
puts " math = [ math::choose 52 2 ]                  "
\$canvas configure -highlightcolor brown -relief raised -border 10
\$canvas configure -width 700 -height 300
# This posting, screenshots, and TCL source code is
# copyrighted under the TCL/TK license terms.
# Editorial rights retained under the TCL/TK license terms
# and will be defended as necessary in court.
# end of file
# gold on TCL club, 5nov2018```

## Comments Section

Please place any comments here, Thanks.

gold 9/27/2021. Switched some comment signs ;# to #. This a big file. Check earlier editions, if not compatible. Maybe obvious, but this page was written on Windows10 Tcl ports including ActiveTCL. I assume that the reader can cut and paste on screen, what the reader needs, and tootle on to his own project and own contribution pages to the TCL Wiki.

 Category Numerical Analysis Category Toys Category Calculator Category Mathematics Category Example Toys and Games Category Games Category Application Category GUI