### Poker Probability and Calculator Demo Example

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## Introduction edit

gold Here is some starter code for poker probability. The impetus for this calculator was checking some poker probabilities of 5 card draw in a book. Its easy enough to plug in formulas into the calculating subroutine. Most of the checked testcases involve 5 card poker.### Testcases Section

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 in the TCL console. Also, some of the pseudocode statements were checked in the google search engine which will take math expressions. The factorial procedure used below was posted by Suchenworth. Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the factorial function.#### Testcase 1

For a system of dealing 5 cards from a 52 card deck, the sample space would be factorial(52)/((factorial(5)*factorial(52-5)) or 52!/(5!*47!). Most of these TCL expressions can be pasted into the console window.set space [ expr { [f 52]/[f 5]/[f 47]} ] ;# generic TCL answer% 2598960The probability of four of a kind would be frequency/ sample space:

set probability [ expr { (1.* 4 )/2598960} ] answer% 1.5391E-6The probability of four of a kind would be frequency/ sample space:

set probability [ expr { (1.* 624 )/2598960} ] answer% 2.40E-4The probability of three of a kind would be frequency/ sample space:

set probability [ expr { (1.* 54912 )/2598960} ] answer% 0.021

#### Testcase 2

Continuing with dealing 5 cards from a 52 card deck, there is an alternate way of estimating the factorial, which is the Swerling formula. The Swerling formula is less accurate for small numbers (<60!), but has some interest in estimating the factorial with large numbers (>60!). The Swerling formula is sometimes used to check the magnitude of large N! in computing. The Swerling formula is implemented and accessible from the report console window.set swerlingspace [ expr { [fact 52]/[fact 5]/[fact 47]} ] answer% 2599403 # and the answers for probabilites would slightly different set probability [ expr { (1.* 4 )/2599403 } ] answer% 1.53881E-6 vis previous 1.5391E-6

#### Testcase 3

There are 55 poker players who wish to sit four at a table. What is the number of combinations? The sample space would be factorial(55)/((factorial(4)*factorial(55-4)) or 55!/(5!*50!). The calculator gives an answer of 341055.#### Testcase 4

Picking up a five card hand, what is the probability and odds of a straight flush, less royal flush ? As caulated before, 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.set straight_flush [expr { (1.*36)/ 2598960 } ] set straight_flush_odds [expr { 1./( (1.*36)/ 2598960 ) } ] answer% 1.39E-5 , 72192.0

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-5 | 72192 |

four of kind | (13 choose 1)* (12 choose 1) *(4 choose 1) | 624 | 624/258960 | 2.4E-4 | 4164 |

full house | (13 choose 1)* (4 choose 3) *(12 choose 2) | 858 | 858/258960 | 2.11E-2 | 46.3 |

three of kind | (13 choose 1)* (4 choose 3) *(12 choose 2)* (4 choose 2) | 156 | 156/258960 | 1.44E-3 | 693 |

#### Testcase 5

Three cards are bought at 100 cents each. Possibly wrongly, but I am told that the chance of receiving a royal card on each draw is "30 percent". The chance of not receiving a royal card on a draw is "70 percent". If all 3 cards are royal, 300 cents are paid;if 2 cards are royal, 150 cents are paid; if one card is royal, 10 cents are paid. What is the weighted pay off or expectation?choose term | term1 * | term2 * | term3 | = wg p(x) | pay off | wg p(x)* pay off |
---|---|---|---|---|---|---|

3 choose 0 | 1.0 | 0.7**3 | 0.3**0 | 0.343 | -100 | -34.3 |

3 choose 1 | 3. | 0.7**2 | 0.3**1 | 0.441 | -90 | -39.89 |

3 choose 2 | 3. | 0.7**1 | 0.3**2 | 0.22 | 50 | 11.0 |

3 choose 3 | 1.0 | 0.7**0 | 0.3**3 | 0.027 | 200 | 5.4 |

sum >> | 1.0 | expectation >> | -57.79 |

weighted probability | pay off | p(x)*pay off |
---|---|---|

.454 | -100 | -45.4 |

0.5322 | -90 | -47.898 |

0.2076 | 50 | +10.38 |

0.027 | 200 | +5.4 |

expectation>> | -77.52 |

### ScreenShot Section:

### References:

- 5-CARD POKER HANDS ,http://www.math.hawaii.edu/~ramsey/Probability/PokerHands.html
- Durango Bill's Poker Probabilities, http://www.durangobill.com/Poker.html
- Poker probability, http://mathforum.org/library/drmath/view/56616.html
- Poker probability, http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Poker_probability.html

#### Appendix Code

#### appendix TCL programs and scripts

# pretty print from autoindent and ased editor # poker probability calculator # written on Windows XP on eTCL # working under TCL version 8.5.6 and eTCL 1.0.1 # gold on TCL WIKI , 12jul2013 package require Tk frame .frame -relief flat -bg aquamarine4 pack .frame -side top -fill y -anchor center set names {{} {number of cards in deck:} } lappend names {distinct hands:} lappend names {frequency: } lappend names {sample space:} lappend names {probability:} lappend names {odds:1} foreach i {1 2 3 4 5 6} { label .frame.label$i -text [lindex $names $i] -anchor e entry .frame.entry$i -width 35 -textvariable side$i grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 } proc about {} { set msg "Calculator for Poker Probability from TCL WIKI, written on eTCL " tk_messageBox -title "About" -message $msg } proc f n { expr {$n < 2 ? 1 : $n * [f [incr n -1]]} } proc fact n {expr { $n<2? 1: $n>20? pow($n,$n)*exp(-$n)*sqrt(2*acos(-1)*$n): wide($n)*[fact [incr n -1]]}} proc calculate { } { global answer2 global side1 side2 side3 side4 side5 global side6 set aa [ expr { int($side1) } ] set bb [ expr { int($side2) } ] set cc [ expr { $aa-$bb } ] set cc [ expr { int($cc) } ] set dd [ expr { int($side3) } ] set term1 [ f $aa ] set term2 [ f $bb ] set term3 [ f $cc ] set samplespace [ expr {$term1*1./(1.*$term2*$term3)} ] set pokerprob [ expr { (1.*$dd)/$samplespace } ] set pokerodds [ expr { 1./$pokerprob } ] set side4 $samplespace set side5 $pokerprob set side6 $pokerodds } proc fillup {aa bb cc dd ee ff} { .frame.entry1 insert 0 "$aa" .frame.entry2 insert 0 "$bb" .frame.entry3 insert 0 "$cc" .frame.entry4 insert 0 "$dd" .frame.entry5 insert 0 "$ee" .frame.entry6 insert 0 "$ff" } proc clearx {} { foreach i {1 2 3 4 5 6 } { .frame.entry$i delete 0 end } } proc reportx {} { global side1 side2 side3 side4 side5 global side6 console show; puts "number of cards: $side1 " puts "distinct hards: $side2 " puts "frequency: $side3 " puts "answer sample space: $side4 " puts "hand probability: $side5 " puts "integer odds:1 $side6 " } frame .buttons -bg aquamarine4 ::ttk::button .calculator -text "Solve" -command { calculate } ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 52. 5. 4. 2.59E6 1.5E-6 6.49E5 } ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 52. 5. 624. 8. 2.4E-4 4164 } ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 52. 5. 54912. 4096. 2.11E-2 47.3 } ::ttk::button .clearallx -text clear -command {clearx } ::ttk::button .about -text about -command about ::ttk::button .cons -text report -command { reportx } ::ttk::button .exit -text exit -command {exit} pack .calculator -in .buttons -side top -padx 10 -pady 5 pack .clearallx .cons .about .exit .test4 .test3 .test2 -side bottom -in .buttons grid .frame .buttons -sticky ns -pady {0 10} . configure -background aquamarine4 -highlightcolor brown -relief raised -border 30 wm title . "Poker Probability Calculator "

For the push buttons, the recommended procedure is push testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console. While the testcases are in units of cards, the units cancel out in the calculator equations.

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