## Let’s Make A Deal!

### July 24, 2009

Since the host always reveals a goat, the only thing that matters is whether the contestant’s initial pick was an auto or a goat; if the initial pick was an auto, staying wins, and if the initial pick was a goat, switching wins. The monty function (named after the original host, Monty Hall), plays n games and reports the number of wins when switching and the number of wins when staying:

```(define (monty n)   (let monty ((n n) (switch 0) (stay 0))     (let ((auto (randint 3)) (pick (randint 3)))       (cond ((zero? n) (values switch stay))             ((= auto pick) (monty (- n 1) switch (+ stay 1)))             (else (monty (- n 1) (+ switch 1) stay))))))```

The variables `switch` and `stay` count the number of wins for each strategy. `(randint `n`)` returns a non-negative integer less than n; variable `auto` records the location of the automobile, and variable `pick` records the location of the contestant’s initial pick.

Running `monty` for 100,000 contests shows a clear winner:

```> (monty 100000) 66824 33176```

You can run the program at http://programmingpraxis.codepad.org/54PDMDSv, including `rand` and `randint`, which are taken from the Standard Prelude.

When the contestant first makes a choice, he wins the automobile one-third of the time. The other two doors hide the automobile the other two-thirds of the time. When the host reveals a goat, the probabilities don’t change: the door first chosen by the contestant wins the automobile one-third of the time, and wins a goat two-thirds of the time. But since the other two doors hid the automobile two-thirds of the time, and we now know that one of the doors certainly does not hide the automobile, the other door must hide the automobile two-thirds of the time. It is better to switch than stay.

When this puzzle appeared in Marilyn vos Savant’s column in Parade Magazine in 1990, it generated more mail than any previous column; over a thousand people with Ph.D.s thought she was wrong.

Pages: 1 2

### 9 Responses to “Let’s Make A Deal!”

1. cwbowron said
```#lang scheme

(define DOOR-COUNT 3)

(define (car? n lst) (eq? (list-ref lst n) 'car))

(define (get-door-layout n)
(let loop ([n n] [doors (list)])
(cond
[(= 0 n) doors]
[(or (member 'car doors)
(not (= 0 (random n))))
(loop (sub1 n) (cons 'goat doors))]
[else
(loop (sub1 n) (cons 'car doors))])))

(define (get-random-door door-count criteria-function)
(let ([door (random door-count)])
(if (criteria-function door)
door
(get-random-door door-count criteria-function))))

(define (get-revealed-door doors contestant-selection)
(get-random-door (length doors)
(lambda (n)
(and (not (= n contestant-selection))
(eq? (list-ref doors n) 'goat)))))

(define (get-switch-door door-count unavailable-doors)
(get-random-door door-count (lambda (n) (not (member n unavailable-doors)))))

(define (make-a-deal n (show-debug-info #f))
(let ([success-count-stick 0]
[success-count-switch 0])
(for ((trial (in-range n)))
(let ([doors (get-door-layout DOOR-COUNT)]
[selected-door (random DOOR-COUNT)])
(let ([revealed-door (get-revealed-door doors selected-door)])
(let ([selected-door-switch (get-switch-door DOOR-COUNT (list selected-door revealed-door))])
(when show-debug-info
(printf "DOORS: ~A~%" doors)
(printf "SELECTED DOOR: ~A~%" selected-door)
(printf "REVEALED DOOR: ~A~%" revealed-door)
(printf "SWITCH DOOR: ~A~%" selected-door-switch))
(cond [(car? selected-door doors)
[(car? selected-door-switch doors)
(printf "Success Rate of Sticking:  ~A%~%" (exact->inexact (* 100 (/ success-count-stick n))))
(printf "Success Rate of Switching: ~A%~%" (exact->inexact (* 100 (/ success-count-switch n))))
(if (> success-count-switch success-count-stick)
(printf "Better off Switching~%")
(printf "Better off Sticking~%"))))

(make-a-deal 100000)
```
2. Jamie Hope said

I already knew that the correct answer is the probability of winning by switching is 2/3, but:

```#!r6rs

(import (rnrs)
(srfi :27))

(define (create-game) ; false = goat, true = car
(let ((v (make-vector 3 #f)))
(vector-set! v (random-integer 3) #t)
v))

(define (player-choice) (random-integer 3))

(define (host-choice g p) ; host always chooses a goat
(case p
((0) (if (vector-ref g 1) 2 1))
((1) (if (vector-ref g 0) 2 0))
((2) (if (vector-ref g 0) 1 0))))

(define (switch p h)
(case (+ p h)
((1) 2)
((2) 1)
((3) 0)))

(define (run-test n)
(let loop ((wins 0) (total 0))
(if (= total n) (/ wins total)
(let* ((g (create-game))
(p (player-choice))
(h (host-choice g p))
(s (switch p h)))
(if (vector-ref g s)
(loop (+ wins 1) (+ total 1))
(loop wins (+ total 1)))))))

(display (inexact (run-test 1000000)))
```

=> 0.66756

3. Mark VandeWettering KF6KYI said

Ugly, but it works.

```#!/usr/bin/env python

# code for the monty hall problem
# written by Mark VandeWettering as a challenge on
# the programming praxis website.

import random

NTRIALS=10000

def trial(switch=False):
# pick a random door...
door = random.randint(1, 3)
guess = random.randint(1, 3)
if not switch:
return door == guess
else:
# slightly clever, but the only way that you
# lose is if you were lucky enough (or unlucky
# enough) to guess the right door in the first
# place.   This gives away the entire problem.
return door != guess

cnt_noswitch = 0
cnt_switch = 0

for x in range(NTRIALS):
if trial():
cnt_noswitch = cnt_noswitch + 1
if trial(switch=True):
cnt_switch = cnt_switch + 1

print "By not switching, you won %d/%d rounds." % (cnt_noswitch, NTRIALS)
print "By switching, you won %d/%d rounds." % (cnt_switch, NTRIALS)
if cnt_switch > cnt_noswitch:
print "It appears you should switch."
else:
print "It appears you should not switch."
```
4. I suggest also reading a nice treatment of this exact problem in “The man who loved only numbers” by Paul Hoffman.

5. Brian Everett said

I’m just now learning Python and figured I’d play around with its list capabilities for this one…

Originally I was using list comprehension to generate a list of random numbers (to indicate which door the car is behind) and keeping the contestant’s choice constant (always picking door number 1), but I realized it might be slightly faster to randomize the contestant’s choice and assume the car is always located behind door number 1 – simply using the list as a looping mechanism.

```import random
trials = 100000
print "Switching will win: %2.2f%% of the time" % (len(filter(lambda trial: random.randint(1, 3) != 1, xrange(trials))) / float(trials) * 100)
```
6. [...] Let’s Make A Deal! « Programming Praxis. [...]

7. Greg R said

I made an attempt in PL/SQL (Oracle). I made the number of doors configurable (assuming that the host opens all doors except yours or the prize, or a random single door if you have chosen the prize door initially).

```declare
Runs  integer := 10000;
Doors integer := 3;

function RunMonteHaul
(
Runs    in  integer,
MaxDoor in  integer,
Swap    in  boolean
)
return integer
is
Prize     integer;
OtherDoor integer;
MyDoor    integer;
Wins      integer := 0;
i integer;
j integer;
begin

for i in 1 .. Runs loop
Prize := dbms_random.value(1,MaxDoor);
MyDoor := dbms_random.value(1,MaxDoor);
if Prize = MyDoor then
if Prize = MaxDoor then
OtherDoor := 1;
else
OtherDoor := MaxDoor;
end if;
else
OtherDoor := Prize;
end if;
if Swap then
MyDoor := OtherDoor;
end if;
if MyDoor = Prize then
Wins := Wins + 1;
end if;
end loop;
return Wins;
end RunMonteHaul;
begin
dbms_output.put_line('Runs  ' || Runs);
dbms_output.put_line('Doors ' || Doors);
dbms_output.put_line('Swap Wins    ' || RunMonteHaul(Runs,Doors,true));
dbms_output.put_line('No Swap Wins ' || RunMonteHaul(Runs,Doors,false));
end;

```

This produces the output

```Runs  10000
Doors 3
Swap Wins    6290
No Swap Wins 3725

Runs  10000
Doors 10000
Swap Wins    9999
No Swap Wins 1
```
8. Diego Giuliani said
```# Monty Hall Paradox simulation / Let's make a deal
# Simulates a number of games and output the total number of wins and looses,
# and the percentage of each, to determine if switching doors after the host has opened one
# is a better option.

from random import random,randint

def scrambleDoors(doors):
"""
Simulates the scramble of doors by choosing one to contain a Car instead of a Goat
"""
doors[randint(1,3)] = "Car"
return doors

def pick():
"""
Choose a door between 1 and 3
"""
return int(random() * 100) % 3 + 1

def openDoor(door_list,door):
"""
The host opens a door, wich must not reveal the price, and also is not the one the player has choosen
"""
return [i for i in door_list.keys() if (door_list[i] != "Car") & (i != door)][0]

def switchDoor(door_list,door):
return [i for i in door_list.keys() if i != door]

def play():
# 1 - Scramble doors
doors = scrambleDoors({1:"Goat",2:"Goat",3:"Goat"})

# 2 - Pick a door
door = pick()

# 3 - Open a door, and remove it from door list
od = openDoor(doors,door)
del doors[od]

# 4 - Switch door
newDoor = switchDoor(doors,door)

# 5 - Won?
return doors[newDoor[0]] == "Car"

def game(n = 100):
"""
n: Number of games to simulate
"""

win = 0
loose = 0
for i in range(0,n):
if (play()):
win +=1
else:
loose += 1
porcWin = (float(win) / n) *  100
porcLoose = (float(loose) / n) *  100
print """
Runs  : % s
Win   : %s  Percentage win  : %s
Loose : %s  Percentage Loose: %s
""" % (n,win,porcWin, loose,porcLoose)
```

Which produces the following output

```Runs  : 1000
Win   : 680  Percentage win  : 68.0
Loose : 320  Percentage Loose: 32.0
```