Updated 2017-03-05 20:56:25 by gold

## 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

The probability of four of a kind would be frequency/ sample space:
```  set probability [ expr { (1.* 4 )/2598960} ]

The probability of four of a kind would be frequency/ sample space:
``` set probability [ expr { (1.* 624 )/2598960} ]

The probability of three of a kind would be frequency/ sample space:
``` set probability [ expr { (1.* 54912 )/2598960} ]

#### 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]} ]
# 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 )   } ]
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

#### 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 offwg 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 5011.0
3 choose 3 1.0 0.7**0 0.3**3 0.027 200 5.4
sum >> 1.0 expectation >> -57.79

Now, if there are 3 royal cards and 10 common cards in a suit, the probability of a royal card is 3/(10+3),3/13, or 0.2307. The probability of not receivng a royal card is 10/(10+3),10/13, or 0.769. Checking, the total chances of either a royal card or a common card are 0.2307 + 0.769 or 1. Since the probabilities are different than first represented, the expectation of pay off should be recalculated.
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

So in buying three cards for three dollars, the expectation is a loss of 78 cents.

### References:

#### 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
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 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 .cons -text report -command { reportx }
::ttk::button .exit -text exit -command {exit}