Back to Nike Missile Home Page

The Three Doors Problem

Related to the "Bertrand's box paradox" of 1889
and Three Prisoners problem, and the Monty Hall problem, all the same logically.

Table of Contents
    - Problem statement
    - Surprising Correct Answer
    - The output of the simulation code agreed with Randy :-((
    - An Approach to the Correct Answer
    - Here is the (FreeBasic) code

Son (Randy) posed the following problem.
Problem statement
The Three Doors (or Monty Hall) Problem
    - a million dollars $ behind one
    - a pile of poop behind one
    - a tiger behind one
You (guest) choose one door
Host
    - opens another door which does not have the million $
    - does not reveal what is behind guest chosen door
You (the guest) get to choose whether to
    a) stay with your original choice
    b) flip the choice to the other closed door
Is plan a) or b) better ?????

Surprising Correct Answer
Randy said that to flip your choice was the best answer.


   I disagreed with his answer.
He was adamant, so I wrote some code (below, with random choices) to prove him wrong.

The output of the simulation code agreed with Randy :-((
   There does not seem a logical error in my code.
Here is the output of the running simulation.
Starting BehindDoors, at 07-06-2017 18:19:14

Guest sticks with original selection
    10000000 Iterations - Guest wins % 33.3319633331963

Guest flips on even iterations
    10000000 Iterations - Guest wins % 50.0038350003835

Guest always flips
    10000000 Iterations - Guest wins % 66.6403366640337

All done!

Terminating BehindDoors, at 07-06-2017 18:19:25

An Approach to the Correct Answer
(Part of an e-mail from Randy - Gina is his wife - )
Gina is happy you also don't understand. (Now she doesn't feel alone.)

But she did say, "well, if there are only two outcomes, you can win or lose, and without switching you lose 33% of the time, then with switching you must win 66% of the time."

Sounds good to me - I even understand it at this overview level :-))


Here is the (FreeBasic) code

' GOAL
'   solve a probability problem
' Background -
'    three doors
'       a million dollars $ behind one
'       a pile of poop behind one
'       a tiger behind one
'    you choose one door
'    host 
'       opens another door which does not have the million $
'       does not reveal what’s behind guest chosen door
'    you (the guest) get to choose whether to
'       a) stay with your original choice
'       b) switch to the other closed door
'  --------------------------
' There is a big uproar about
'     is it better to stay with original choice
'     flip choice to other closed door, under which conditions
'
'
' Plan of Attack
'   using the language supplied random number generator,
'    a) set the three items behind random doors
'    b) have guest choose a door
'    c) have host open a non-winning door
'   say 10,000 times
'  Use successive phases to 
'      1) stick with guest door - probability of winning
'      2) random have guest change selected door
'      3) change selected door to non-shown door
' ------------
'  The Report we  want to make is
'    a) assurance of probability (random number generator) working
'    b) statistics of each each successive phase, including
'       - % which behind which door   
'       - % which door initial choice
'       - % win
'
' Format of report
'" date/time phase #
'    phase rules (1,2,3  above)
'  Summary
'  % $ behind which door
'  % guess chooses which door
'  % guest win, poop, tiger
'  Details of last 5? sequences, (in "Plan of Attack")
'     a) what behind each door
'     b) guest 1st choice
'     c) host shown door
'     d) guest final choice
'
'-------------------  
' set monitor to 1280 x 1024
'  adapted to write and read the Accumulator array
'  %  integer   +-32K
'  &  long integer  +-2 billion
'  !  single prcision float   7 places*10^+-38
'  #  double precision float 15 places*10^+-300
'  $  st-ring - bug - error report on correct spelling?
'
' lets do it specifically, not generally say with n doors ;-))
' so - array of three doors

Dim shared Phase as integer ' 1 thru  3, used to determine what the guest does
dim shared Iteration as integer
dim shared RandInt as integer  ' random integer between 1 and 3

dim shared x, y, z as integer  ' working variables

DIM Shared BehindDoor(3) as Integer ' values behind doors
Const Tiger = 1
Const Poop = 2
Const Million = 3
Const debug = 0
DIM Shared GuestChoice as Integer 
DIM Shared GuestFlip as Integer 
DIM Shared HostChoice as Integer 
dim shared GuestWins as Integer
dim shared GuestTrys as Integer

'
' Statistical Data Fields
dim shared SDoorTotals(3) as integer
dim shared SDoorTiger(3) as integer
dim shared SDoorPoop(3) as integer
dim shared SDoorMillion(3) as integer


 


 'DEFDBL A-Z         ' default everything to double float, not DEFSNG
 'DEFSNG A-Z        ' default everything to single float, not DEFDBL - AH - color!!
'   well, it fails on DEFSNG - what is going on?? - what needs double? Trig??
Option Explicit 
' Constants 

 'transparincies
CONST White = &hFFFFFF  ' colors
CONST Grey  = &h606060
CONST GreyL = &hA0A0A0
CONST Red   = &hFF0000
CONST RedDull = &h800000
CONST Yellow= &hFFFF00
CONST YellowDull = &h808000
CONST Green = &h00FF00
CONST Blue  = &h0000FF
CONST BlueDull = &h000080
CONST Black = &h000000
 
' this subroutine is used because occasionally
'    RandInt is returned as zero
' Goal, return a random integer value, RandInt, between 1 through 3
Sub  FindRandInt   
   RandInt = 0
    while ((RandInt < 1) or (RandInt > 3))
      RandInt = (rnd(1)*3)+0.5 ' get a
    WEnd
End Sub

Sub XFlip
    dim yy as integer
    Select Case GuestChoice
    Case 1
        if HostChoice = 2 then 
            yy =3
        else 
            yy = 2
        endif
    Case 2
        if HostChoice = 1 then
            yy =3
        else
            yy = 1
        endif
     Case 3
        if HostChoice = 1 then
            yy =2
        else
            yy = 1
        endif
     End Select
     GuestChoice = yy
End Sub
    
    


 
  
SCREEN 21, 32, 1,0 ' 15 ' 14  ' 12 '13
    ' 16-512x384; 17-640x400; 18-640x480; 19-800x600; 20-1024x768; 21-1280x1024
 Randomize timer
    
 print ""   
print "The Three Doors Problem"
print "     - a million dollars $ behind one"
print "     - a pile of poop behind one"
print "     - a people eating tiger behind one"
print "    You (guest) choose one door"
print "    Host "
print "     - opens another door which does not have the million $"
print "     - does not reveal what is behind guest chosen door"
print "    You (the guest) get to choose whether to"
print "       a) stay with your original choice"
print "       b) flip the choice to the other closed door"
print "Is plan a) or b) better ?????"
print "

Open "C:\FreeBasic\3Doors.txt" for Output as #1
 print #1,""   
print #1,"The Three Doors Problem"
print #1,"     - a million dollars $ behind one"
print #1,"     - a pile of poop behind one"
print #1,"     - a people eating tiger behind one"
print #1,"    You (guest) choose one door"
print #1,"    Host "
print #1,"     - opens another door which does not have the million $"
print #1,"     - does not reveal what is behind guest chosen door"
print #1,"    You (the guest) get to choose whether to"
print #1,"       a) stay with your original choice"
print #1,"       b) flip the choice to the other closed door"
print #1,"Is plan a) or b) better ?????"
print #1,"
 '----------------------------------
    Print "press any key to continue"

WHILE INKEY$ = "": sleep (10):WEND

 Print ""
  print "Starting BehindDoors, at  ";date$,time$()
Print ""
 Print #1,""
  print #1,"Starting BehindDoors, at  ";date$,time$()
Print #1,""
   
    
Phase = 1
while (Phase < 4) ' see meaning of Phase above
    Select Case Phase
    Case 1
        print "Guest sticks with original selection"
        print #1,"Guest sticks with original selection"
    Case 2
        Print "Guest   flips on even iterations"
        Print #1,"Guest   flips on even iterations"
    Case 3
        Print "Guest always flips"
        Print #1,"Guest always flips"
    Case else
        Print "XXXXXXXXX bad case"
    End Select
    ' clear statistics counters
    
    for x = 1 to 3  
        SDoorTotals(x) = 0
        SDoorTiger(x) = 0
        SDoorPoop(x) = 0
        SDoorMillion(x) = 0
    next x
    GuestWins = 0
    GuestTrys = 0
    '--------------
    Iteration = 1
    While (Iteration < 10000000)  
        FindRandInt
        BehindDoor(1) =  RandInt    
        FindRandInt
        BehindDoor(2) =  RandInt   ' *** a subroutine to set a 0 < random int < 4
        FindRandInt
        BehindDoor(3) =   RandInt   ' *** a subroutine to set a 0 < random int < 4
        While (BehindDoor(1) = BehindDoor(2))    ' get a different value into BehindDoor2
            FindRandInt
            BehindDoor(2) =   RandInt   ' *** a subroutine to set a 0 < random int < 4
       WEnd 
       While ( (BehindDoor(3) = BehindDoor(1)) or (BehindDoor(3) = BehindDoor(2)))
            FindRandInt
            BehindDoor(3) =   RandInt   ' *** a subroutine to set a 0 < random int < 4
        WEnd
        ' we now have three unequal values behind all three doors
        '
        'OK, we have the doors set up, lets do the guest selection
        FindRandInt ' new random #
        GuestChoice = RandInt  ' guest chooses this (random) door
        ' now host has to choose guest door not and million door not
        FindRandInt ' new random #
        while ( (GuestChoice = RandInt) or (BehindDoor(RandInt) = Million ))
            FindRandInt
        WEnd
        HostChoice = RandInt  ' ok, we have the door that host opens
        ' now compute the door for GuestFlip
        '    rules, not the door the host opens
        
        if (Iteration<6 and debug) then
            Print "Iteration=";Iteration,BehindDoor(1),BehindDoor(2),BehindDoor(3) 
            print "GuestChoice =";GuestChoice,"HostChoice = ";HostChoice,
            if (BehindDoor(GuestChoice) = Million) then
                print "Guest wins"
            else
                print "XXXX"
            Endif
        EndIf
        Select Case Phase
        case 1 ' dont flip
        Case 2 ' flip on even
            if NOT ( Iteration and 1) then
                XFlip
            endif
        Case 3 ' always flip
            XFlip
        End Select
        
        if (BehindDoor(GuestChoice) = Million) then
            GuestWins += 1
        Endif
       ' GuestTrys += 1
        Iteration += 1
      wend
    print Iteration;" Iterations ",
    print " - Guest wins % ";(100*GuestWins)/Iteration
    Print ""
    print #1,Iteration;" Iterations ",
    print #1," - Guest wins % ";(100*GuestWins)/Iteration
    Print #1,""
    Phase += 1
wend  
   PRINT "All done!"
   PRINT #1,"All done!"
    Print ""
  print "Terminating BehindDoors, at  ";date$,time$()
Print ""
 Print #1,""
  print #1,"Terminating BehindDoors, at  ";date$,time$()
Print #1,""

   Close #1
   
print "press any key to terminate"
WHILE INKEY$ = "": sleep (10):WEND
END  ' kills text screen


Started July 13, 2017
Last update July 13, 2017
Back to Nike Missile Home Page