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

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

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."

' 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