Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I just learned about the Monty Hall problem and found it quite amazing. So I thought about extending the problem a bit to understand more about it.


In this modification of the Monty Hall Problem, instead of three doors, we have four (or maybe $n$) doors, one with a car and the other three (or $n-1$) with a goat each (I want the car).

We need to choose any one of the doors. After we have chosen the door, Monty deliberately reveals one of the doors that has a goat and asks us if we wish to change our choice.

So should we switch the door we have chosen, or does it not matter if we switch or stay with our choice?

It would be even better if we knew the probability of winning upon switching given that Monty opens $k$ doors.

share|improve this question
21  
As a side note, if you're trying to change the problem for the purpose of teaching it, (rather than just creating a fun variety,) you should have Monty reveal n-2 goats, where n=total doors at start. That still matches the original problem (he'd open 1 of the 3,) and it makes the correct answer (not switching) really obvious as N increases. "You pick 1 of 100 doors. He opens 98, revealing goats. Do you keep the one you picked, or switch to the other?" (See @Tony's answer below - I didn't before posting this, but it makes the same point. +1 to him.) –  Jaydles Dec 16 '13 at 17:38
19  
OMG that is the perfect monty hall... I have a 3 in 4 chance of finally getting the goat my wife keeps telling me I cant have! –  Chad Dec 16 '13 at 23:01
4  
@Jaydles I don't know, people's intuitions are different. Two years ago, I saw that exact explanation, with a million doors rather than 100 (making it even more "obvious"). I still thought switching made no difference. –  user54609 Dec 17 '13 at 1:15
5  
A visual explanation: http://blog.vctr.me/monty-hall. Scroll down to the bottom to play with the strategy/doors/speed :) –  thgaskell Dec 17 '13 at 3:16
5  
@einpoklum - The car is still the better outcome. If nothing else, it can be sold or traded for many goats... Enough to create a self-sustaining herd given sufficient suitable land. :) –  James Snell Dec 17 '13 at 12:01

20 Answers 20

up vote 90 down vote accepted

I decided to make an answer out of my comment, just for the heck of it.

Suppose we have $n$ doors, with a car behind $1$ of them. The probability of choosing the door with the car behind it on your first pick, is $\frac{1}{n}$.

Monty then opens $k$ doors, where $0\leq k\leq n-2$ (he has to leave your original door and at least one other door closed).

The probability of picking the car if you choose a different door, is the chance of not having picked the car, which is $\frac{n-1}{n}$, times the probability of picking it now, which is $\frac{1}{n-k-1}$. This gives us a total probability of $$ \frac{n-1}{n}\cdot \frac{1}{n-k-1} = \frac{1}{n} \cdot \frac{n-1}{n-k-1} \geq \frac{1}{n} $$ If Monty opens no doors, $k = 0$ and that reduces to $\frac{1}{n}$.

For all $k > 0$, $\frac{n-1}{n-k-1} > 1$ and so the probabilty of picking the car on your second guess is greater than $\frac{1}{n}$.

If $k$ is at its maximum value of $n-2$, the probability of picking a car after switching becomes $$\frac{1}{n}\cdot \frac{n-1}{n-(n-2)-1} = \frac{1}{n}\cdot \frac{n-1}{1} = \frac{n-1}{n}$$For $n=3$, this is the solution to the original Monty Hall problem.

Switch.

share|improve this answer
8  
I like the added k, seems like the best answer so far (at least for me) –  Timo Huovinen Dec 16 '13 at 21:05
    
If monty opens all doors except for one, then k will be n-2. Putting it in the general formula you made, the probability is $\frac{n-1}{n}$ . As n gets larger, the probability that we win increases. Infact, if number of doors is 40, a simulation on blog.vctr.me/monty-hall shows that we win 99 out of 100 times. Amazing! –  shaurya gupta Dec 17 '13 at 14:00
    
Although I had accepted the answer by albramo, I accepted yours. The $k$ man, the $k $ ! –  shaurya gupta Dec 17 '13 at 14:02
2  
Wow, thanks. I actually based my answer on a comment I made on Abramo's answer, so I kind of based it on that answer. –  SQB Dec 17 '13 at 15:32
    
+1. The addition of $k$ makes this a really nice extension of the original problem. –  Tom Chantler Dec 18 '13 at 11:10

I do realize this is the math site but I'm not actually very good at math. I do enjoy a brain teaser like this though and also enjoy trying to make it easy for people to understand. Here is how I think of this problem.

If you get 100 chances to take this test you will choose correctly 33 times (1 out of 3) and incorrectly 67 times. By switching you win 67 out of 100 times. Simple.

As others have noted, for four doors the odds are even better for switching if you eliminate all but two doors. Your initial guess will be right 25% of the time and wrong 75% of the time.

And the odds only get better as the number of doors increases... :)

share|improve this answer

The typical answer to the Monty Hall question is ridiculous.

The typical argument is:

  1. You had 1 in 3 of choosing corretly.
  2. Monty opens a door. This will never be the correct door.
  3. You now have a 1 in 2 of choosing correctly so switch.

I've never understood the reasoning behind this because with the removal of the incorrect door from the equation, your original choice has changed from 1 in 3 to 1 in 2; because the correct door is either the door you already selected or the other door. The fact that the original choice was 1 in 3 no longer matters; Monty reduced the problem domain.

At this point in time, you have 50% chance that your original door is correct and 50% chance that the other door is correct.

This is not a compelling reason to change.

Basically:

  1. You start with 33% chance of any door being correct.
  2. Monty removed a door.
  3. Now the probability of any door being correct has increased to 50%; both doors are equally correct.

Response to many (in my opinion goofball) comments:

Short form: LRN2MATHSN00Bz

Longer form: All the responses that say sticking is 1/3 chance to win and changing is 2/3 chance to lose are, as noted above, ridiculous.

The domain set starts at 3 doors. While the domain set remains unchanged, the choice of any one door has a 1/3 chance of being correct and a 2/3 chance of being worng.

As soon as Monty removes a door, the domain set becomes 2 doors; this increases your original selection to a 50% winner.

The classic argument of you're won doar choice remains 1/3 chanse to winz is (as I attempted to demonstrate with interesting spellings) entirely incorrect. The revealed door does not just increase the chance of the other door wining, it also increases the chance of your current door winning equally.

share|improve this answer
    
This is the problem with probability in general. When the final choice is made it has become a 50/50 probability for that moment in time between desired outcome and undesired outcome. Convergence to the mean essentially makes all probability data worthless in the grand scheme of things... but these problems make for some decent dinner-party talking points (in the right company). Edit: The down votes you received are both uncalled for and confusing. –  Mike Hometchko Dec 17 '13 at 21:23
9  
The down votes he received are because he's wrong. Just look at some of the other examples (like the ones with 100 doors). You had a 33% chance of being correct in your initial pick. Monty removing a door cannot change your original probability retroactively. If you were to run this experiment 100 times, NOT switching doors will give you a win about 33% of the time (try it!). Switching will win about 50%. –  Gerrat Dec 17 '13 at 22:54
5  
Gerrat is not correct. Another argument to intuit the right answer is this; say you and a friend play this game. You never switch, and your friend always does. You both always pick the same initial door. You have a 1/3 chance of winning, always. Since Monty ALWAYS picks a losing door, and your friend always switches, you now have collectively picked both remaining doors, and Monty has shown you a losing door. So ONE of you is ALWAYS going to win. Since you are always going to win 1/3 of the time, your friend is going to win the other 2/3 of the time. –  Michael Campbell Dec 18 '13 at 1:21
1  
@DwB the choice has changed from 1 in 3 with even probability, to 1 in 2 with uneven probability. So you're right about 1 in 2, but it doesn't translate in 50%. I wrote a different explanation here that might help: math.stackexchange.com/a/611881/116229 –  Christophe Dec 18 '13 at 19:05
1  
@MichaelCampbell Thanks for pointing that out I didn't notice that Gerrat said you would win 50% of the times by changing. Anybody who doubts that you win 2/3 of the time by switching and win 1/3 of the time by not switching needs to sit down with a piece of paper and explicitly construct the sample space (takes about 10 minutes). Of all the possible outcomes when always switching 2/3 them win. If that is not convincing enough get some adequately random numbers and write a simple computer program to simulate the game and look at the outcomes when the player always switches. –  Spencer Dec 19 '13 at 2:19

Made an account just to answer this. General feeling seems to be that switching your selection increases the odds because you make a new selection from smaller subgroup, and combined odds are greater than initial 1/n.

This, however, is just an illusion. By NOT switching, you are still making a new selection. And your initial selection belongs always the the reduced subgroup (as Monty never opens the window you selected).

Probability to select right one is always 1/(n-k), where k = windows Monty will open, 0 >= k <= n-1.

The initial selection is not meaningful, and that phase can be skipped. You are only selecting one windows that belongs to the reduced subgroup. All that matters is what you select in the later one. Good news is that by switching you never reduce your odds either.

Edit: Just to clarify a bit. Common mistake seems to be assuming that if you do NOT switch, you are still choosing from 1/n. This is not the case, you are just re-selecting your old answer and choosing from 1/(n-k).

Edit 2: After reading responses and reconsidering this, I must concede. Switching your answer does increase the odds. Extreme case got me convinced: million doors, make selection, Monty removes all but 2. Now your first door has next to none odds of being right, but the one other left is almost certainly.

share|improve this answer
    
I doubt you could code a simulation of this which (a) follows the question and (b) gives your result. –  Henry Dec 18 '13 at 10:57
    
Thanks! I'd upvote you but not enough points yet :) –  Sopuli Dec 18 '13 at 10:57
    
I've just reevaluated myself all the facts about the problem, and now I'd like to cancel what I just said. Everytime I get to this problem I fall in the same pitfall: when considering the probabilities of the second pick, I always forget to apply conditional corrections that come with new knowledge obtained from opening X windows. Then, the core of the problem looks more complicated, and again related to 'common sense' and 'natural language expressiveness'. Sopuli: double, or triple-check what probability you are talking/thinking about. There's where your error lies! –  quetzalcoatl Dec 18 '13 at 11:09
    
At the final point of the game, the probability that the prize is at option A and probability that the prize is at option B is NOT 50%/50%. It would be 50-50 if we had NO KNOWLEDGE of the past events. At the start of the game, the window-prize setup was assumed to be perfectly-flat, because we knew nothing. Hence, for 3 options, 1/3 1/3 and 1/3. The issue is that Monty not eliminate options at random, and this causes the distribution to cease to be flat. –  quetzalcoatl Dec 18 '13 at 11:18
    
He NEVER opens the winning window is very important. He knows something we don't. He never opens the door you picked, and he never removes the winning one. 3 doors, one prize. At first you pick door A. BC are left. Now, if A was winning (1/3), Monty will drop one of B or C at random so both have (1/2) of being dropped. If B was winning (1/3), Monty will always DROP C(1/1). If C was winning (1/3), Monty will always DROP B(1/1). Now, write down the final probs for each case. –  quetzalcoatl Dec 18 '13 at 11:26

My twist on explaining the Monty Hall problem:-

You pick one door.

Monty, instead of revealing a goat behind one of the other doors, offers you BOTH of them if you switch from your initial choice. Of course you switch, since this will absolutely double your chances.

And of course, at least one of the two doors that you switch to must have a goat, Monty just didn't show it to you. So what if he does or doesn't show it? We all know that it's there.

Choice wise, this is exactly the same as the original problem. The illusion is that you don't get the door that Monty opens. But effectively you do, it's just that you disregard it because it has no value.

share|improve this answer

@DwB

Your explanation would work if you chose your door AFTER you were shown a door that had a goat behind it.

Since you make your choice while there are 3 choices, the probability you choose the right door is 1/3 no matter if he shows you a goat or not.

If you choose one of the goats at first, you have a probability of 2/3 of choosing one of those right? So the beauty of this problem is that if you choose a goat first, and switch you will always get the car.

Think about it, if you choose one of the goats first, then the other goat will always get shown. If you switch then you will always switch to the car.

You want to choose a goat first to win this game by SWITCHING.(2/3) You want to choose a car fiRst to win this game by NOT SWITCHING.(1/3)

4 doors...

You choose a car at first(1/4) You choose a goat at first(3/4)

He opens 2 doors with a goat behind them, so your first choice and the last door is left.

If you chose the car first and switched now, you would lose.

If you chose any of the 3 goats initially, and now switched, you would win the car.

It just comes down to the fact you have a better chance of picking one of the goats initially, so assuming he shows all the other "bad choices" to you, it is better to switch your answer you chose originally.

The probabilities you win by switching your choice, assuming that Monty opens all the other doors accept for the original one you picked and one other one.

3 doors (2/3) 4 doors (3/4) 5 doors (4/5) 6 doors (5/6) 7 doors (6/7) 8 doors (7/8)

n doors [(n-1)/n] = 1 - 1/n

as n approaches infinity 1/n becomes 0, and the general probability = 1 - 0 = 1.

So if we were to play this game with an infinite amount of doors, and switch our choice, we would win 100% of the time!

share|improve this answer

The Monty Hall problem always confused me. And so, I coded it up in Matlab (2013a) to better understand it. In this version, the number of games played and N, the number of doors per game, are user settable parameters. I found 100,000 games gives relatively stable win/loss distributions for N = 100.

Code:

%Monty Hall problem assuming N doors, and N-2 doors
%opened prior to the final door choice.

clear;clc
rng('shuffle');

%    USER INPUT PARAMETERS
%------------------------------------------------------------
games = 100000;     %number of Monty Hall games (experiments)
N = 100;            %number of doors in each Monty Hall game
%------------------------------------------------------------

always_losses = 0;  %number of losses for always switch strategy
always_wins = 0;    %number of wins for always switch strategy
never_losses = 0;   %number of losses for never switch strategy
never_wins = 0;     %number of wins for never switch strategy

for ii = 1:games

    prize = randi(N);
    initial_choice = randi(N);

    %(N-2 doors opened prior to final choice)
    if initial_choice == prize
        always_losses = always_losses + 1;  %always switch strategy
        never_wins = never_wins + 1;        %never switch strategy
    else
        always_wins = always_wins + 1;      %always switch strategy
        never_losses = never_losses + 1;    %never switch strategy
    end

end

fprintf('\nWin fraction -- never switch:  %4.3f',never_wins/games);
fprintf('\nWin fraction -- always switch:  %4.3f\n\n',always_wins/games);

Results (for games = 100,000 and N = 100):

Win fraction -- never switch:  0.010
Win fraction -- always switch:  0.990

The computational results support Abramo's observation, namely, that Monty's help is not all that useful as the number of doors increases: The probability of winning approaches 1.0 for the "always switch" strategy as the number of doors approaches infinity.

Letting N = 3 reduces to the familiar Monty Hall game, and, produces the correct answer for that case.

share|improve this answer

The correct answer is, the problem isn't defined clearly enough to arrive at the "correct answer." Let's ignore the n doors, and k doors revealed for a moment, because they aren't essential for realizing that we don't know enough information to solve the question.

We need to choose any one of the doors. After we have chosen the door, Monty deliberately reveals one of the doors that has a goat and asks us if we wish to change our choice.

So should we switch the door we have chosen, or does it not matter if we switch or stay with our choice?

One of the problems with this question, is the motivations of the host are unknown. You specifically said that he revealed one of the doors, but not whether:

  • He always reveals a door.

  • He always reveals a losing door.

Without knowing that information, it is impossible to know if you should switch, not switch, or it doesn't matter.

share|improve this answer

Let's reason in the case of $n$ windows and just $1$ car.

When Monty gives you the opportunity to modify your initial choice, it is definitely better to take it. Here's a way to see it:

The probability of finding the car with the first choice is just $\frac{1}{n}$, and this is the same probability that you have of winning if you keep your initial choice.

Of course you will win by changing your choice if the intersection of two events will happen:

  1. You didn't choose the car with the first choice ( $P(A) = \frac{n-1}{n}$ )

  2. You are lucky and you find the car with your second choice ( $P(B\mid A) = \frac{1}{n-2}$ )

So the probability of winning by changing your mind is the product $$ P(A\cap B) = P(A)\cdot P( B\mid A ) = \frac{n-1}{n}\cdot \frac{1}{n-2} = \frac{n-1}{n(n-2)} > \frac{1}{n} $$ Since $ \frac{n-1}{n-2} \to 1 $ when $n\to \infty$, we notice that Monty's help is not that useful in the case of many many windows.

share|improve this answer
3  
Wow. This is beautiful. –  Cruncher Dec 16 '13 at 14:58
    
We can generalize even further and have Monty opening $k$ doors (with $k <= n - 2$), for a probability of picking the right door on your second guess of $\frac{1}{n - k - 1}$. –  SQB Dec 16 '13 at 19:19
    
To add to this answer a little bit, consider the case with $n$ doors where Monty opens each of the remaining doors except for 1 (another, different generalization of the original problem to OP's). Then, again, if you don't switch your probability of winning is $\frac{1}{n}$, and if you do switch, your probability of winning is exactly equal to the probability that you chose incorrectly at first (this is identical to the case with 3 doors). Then your probability of winning is $\frac{n-1}{n}\rightarrow 1$ as $n\to\infty$, so in this case, Monty's help is incredibly useful. –  crf Dec 16 '13 at 22:30
13  
"The above two events are independent" - no they are not. It is obvious that if you chose the car on your first pick, you will get a goat on your second. You need to rephrase this in terms of the conditional probability that you will choose right on your second choice, given that you chose wrong on your first. –  user2357112 Dec 16 '13 at 23:15
2  
@shauryagupta This is how I finally convinced my dad about this problem. I first said, we'll play the game with cards. I'll know where the ace is, and I'll ask you to pick and if you want to swap. We did about 6 times, but you don't reach convincing results fast enough. So I said, scrap that. We'll use the whole deck. You need to pick the ace of spades. After he picked one, I took the rest of the deck away except for one card. Then asked if he wanted to swap. It was pretty obvious at this point. –  Cruncher Dec 17 '13 at 16:43

You have 1/3 of changes of picking the car. If you pick the car and you accept to change the door, you lose the car.

You have 2/3 of chances of picking the goat. If you pick the goat and you accept to switch the door, you get the car.

So you have 2/3 of chances of getting the car if you switch the door.

All this assuming that the host always opens the door with a goat, and ignoring the psychological factor. If you're able to read the host's mind because you have good psychological skills, then you shouldn't follow the statistics.

share|improve this answer

To add a bit to this discussion, there is a good book, "The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser" by Jason Rosenhouse (published in 2009), which covers all these variants and includes a variety of explanations for understanding them. The book also gives a nice history of the problem. Unfortunately it does not include the xkcd cartoon.

share|improve this answer

Along the same lines of interesting Monty Hall extensions, Rosenthal goes over a couple in this paper http://probability.ca/jeff/writing/montyfall.pdf

share|improve this answer

Last time I came across a Monty Hall question I wrote my own tester in Java. Here it is - with the number of doors set to 4:

public class MontyHall {

    // How many doors.
    static final int Doors = 4;

    // Prizes.
    enum Prize {

        Goat,
        Car;
    }

    public static void main(String[] args) throws InterruptedException {
        // The number of times we would have won if we switched.
        int winWhenSwitched = 0;
        // The number of times we would have won if we didn't switch.
        int winWhenNotSwitched = 0;
        // The number of times we would have lost both ways.
        int lost = 0;

        // n doors.
        Prize doors[] = new Prize[Doors];
        // N Tests
        for (int i = 0; i < 100000; i++) {
            // Put a car behind just one door.
            pickRandomDoorForCar(doors);
            // Make my first choice.
            int firstChoice = randomDoor();
            // Open one remaining goat door. 
            int goatDoorOpened = openOneGoatDoor(doors, firstChoice);
            // What would have been my second choice - if I switched?
            int secondChoice = makeSecondChoice(doors, firstChoice, goatDoorOpened);
            // Count wins/losses.
            if (doors[firstChoice] == Prize.Car) {
                // We would have won without switching!
                winWhenNotSwitched += 1;
            } else {
                // We win if we switched to the car!
                if (doors[secondChoice] == Prize.Car) {
                    // We picked right!
                    winWhenSwitched += 1;
                } else {
                    // Bad choice.
                    lost += 1;
                }
            }
        }
        System.out.println("Wins when switched = " + winWhenSwitched);
        System.out.println("Wins when not switched = " + winWhenNotSwitched);
        System.out.println("Lost = " + lost);
        System.out.println("Proportion = " + ((double) winWhenSwitched / (double) winWhenNotSwitched));
    }

    // Open one door exposing a Goat.
    private static int openOneGoatDoor(Prize doors[], int firstChoice) {
        for (int j = 0; j < Doors; j++) {
            // Not the one already picked and must contain goat.
            if (j != firstChoice && doors[j] == Prize.Goat) {
                // Can only be one of them.
                return j;
            }
        }
        // Should never get here - TODO - Throw an exception here.
        return -1;
    }

    // Make a second choice - avoid first choice and opened door.
    private static int makeSecondChoice(Prize[] doors, int firstChoice, int goatDoorOpened) {
        int secondChoice = randomDoor();
        while (secondChoice == firstChoice || secondChoice == goatDoorOpened) {
            // Try again.
            secondChoice = randomDoor();
        }
        return secondChoice;
    }

    // Pick a random door and put a Car there.
    private static void pickRandomDoorForCar(Prize[] doors) {
        // Start all goats.
        Arrays.fill(doors, Prize.Goat);
        // Pick a random for the car.
        doors[randomDoor()] = Prize.Car;
    }
    // Start my random number generator.
    static final Random random = new Random(System.currentTimeMillis());

    // Pick a random door.
    private static int randomDoor() {
        return random.nextInt(Doors);
    }
}

The results print:

Wins when switched = 37483
Wins when not switched = 24888
Proportion = 1.5060671809707489

I hope this is helpful.

share|improve this answer
    
You beat me to it. –  topher Dec 17 '13 at 19:03

By not switching, you win a car if and only if you chose correctly initially. This happens with probability $\frac{1}{4}$. If you switch, you win a car if and only if you chose incorrectly initially, and then of the remaining two doors, you choose correctly. This happens with probability $\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}$. So if you choose to switch, you are more likely to win a car than if you do not switch.

You never told me whether you'd prefer to win a car or a goat though, so I can't tell you what to do.

xkcd #1282

share|improve this answer
43  
Credit to the origin of the picture xkcd.com/1282 –  rschuler Dec 16 '13 at 15:00
8  
This is one of the best explanations of this I have ever read. –  Jakob Dec 17 '13 at 7:42
1  
I once lived out in the country, and it was a dream of mine to get two goats to keep the grass short. I wanted to name one of the goats Briggs, and the other one Stratton. –  J.R. Dec 17 '13 at 10:49
3  
Goats are browsers; they'd prefer it if your yard was full of trees, bushes, brambles, and stuff they can reach up to. Although they'll graze if that's what's available they do better and stay healthier on browse. –  Bob Jarvis Dec 17 '13 at 19:12
1  
@mousomer: one of our funniest experiences was transporting a goat kid home in the back of a Subaru. Driving up I-77 we got a lot of really odd looks from people. (As one car passed us the woman in the passenger seat looked at us and I clearly saw her say, "What the f...?" :-). Don't think I ever laughed that much while driving, before or since - an' I swear, I was sober as a judge..! :-) –  Bob Jarvis Dec 18 '13 at 16:55

Another way to explain it based on your "gut feeling".

Although this is the "psychological" way of viewing it (so no hard proof like already given above).

Imagine there were 100 doors, one prize is behind one of the doors. Now pick one door. Monty knows behind which of the doors the prize is and now opens all doors except two: your door and another one. Now you get the option to switch, your original door or the one door left closed... You see where I am going?

If you picked, say, door 15. Then all doors go open except 15 and e.g. 63. Doesn't that make 63 really suspicious for containing the prize, whereas your door remained closed only because you picked it initially?

I think this is a beautiful way to "explain" this problem without getting into the math too much. This way, you can convince people of the better option: switch, because you're getting better information. If you then explain the math behind it, then it will become an easy problem!

share|improve this answer
1  
I like to use picking the ace of spades from a deck as an example. Pick any card. I then turn over 50 cards, none of which are the ace of spades. Do you think I have the ace of spades, or do you think you have it? –  Kyle Hale Dec 18 '13 at 16:45

I think that all those trying to show how your chances get progressively better don't see the forrest because of the trees. Take the following:

0) The only thing that matters / gives the probability is the number of doors when you have to make the last choice (classic scenario : 1 out of 3 eliminated, 2 remaining, 50%/50% chance). See "hints" at the bottom.

1) If you add one meaningless door to be eliminated, after the elimination you are in the same situation as in point 0)

2) (a.k.a 1 + 1) If you add two meaningless doors, after eliminating one, proceed like in point 1

3..+inf) by induction, adding whatever the number of doors doesnt change the resulting dilemma / probability.

HINTS: Sticking to the door you've chosen previously is the same as choosing another, it is a 1:N choice.

Comparing "chances" between "rounds" is what muddies the waters, and all the choices except the last are meaningless (or: opening of the doors reveals no interesting information).

share|improve this answer
11  
Big mistake here! The eliminated door is meaningful, because Monty doesn't make a random choice. –  Christophe Dec 16 '13 at 21:39
    
It is meaningless from the point of view that you allways come to an end-game where you choose one out of N. You know the probability in advance, my point is that sticking to the same door is as much of a choice as choosing different ones. Or, that as much as the "probability of other doors rises", so does the "probability of the doors you already chosen" ... –  Miloslav.Raus Dec 16 '13 at 22:34
3  
Hell, now I've convinced myself that from other point of view the risk of picking wrong door indeed does decrease with the number of doors. Anyways, @OldCurmudgeon inspired me with his idea. I'm gonno go code my own (to the point i avoided going through his code) Monte Carlo sim, run an insane number of iterations and report back. –  Miloslav.Raus Dec 17 '13 at 1:50
12  
No Monty Hall question is complete without someone disagreeing with the (well established) answer. –  Joe Dec 17 '13 at 4:30
1  
I doubt you'll need to run an "insane" number of iterations; 30 or 40 should do the trick. Just make sure that the revealed door never has the grand prize. –  J.R. Dec 17 '13 at 10:52

The paradox changes only slightly when you increase the number of doors. The three-door model is most popular because it's the simplest statement of the problem (and the one normally seen on "Let's Make A Deal").

Basically, the reasoning behind the MHP is that, with $N$ doors, your original choice has a $\dfrac{1}{N}$ chance of being the car, which is relatively small. That means the chance that the car is elsewhere is $\dfrac{N-1}{N}$ (or $1 - \dfrac{1}{N}$, same thing). By opening one of these doors to reveal a goat, you don't change the math in the way you expect; the chance that you didn't pick the car (and thus it's behind another unopened door) is still N-1/N, but now that probability is evenly distributed between N-2 unopened doors, so the chance of any one unopened, unchosen door having the car is $\dfrac{N-1}{N(N-2)}$, which will be greater than $\dfrac{1}{N}$ for all $N > 2$.

So, with four doors, the probability that your original pick was correct is .25 ($1/4$), meaning the probability that you didn't pick the car (and that it's behind one of the three other doors) is .75 ($3/4$). Open a door to reveal a goat, and now, there is that same 75% chance that the car is behind either of the two remaining unchosen doors, so switching to either of them gives you a 37.5% chance of choosing correctly ($3/(4(2)) = 3/8$), while staying with your original choice gives you only the same 25% chance.

As you can see, the advantage (the percentage increase in probability of winning) starts shrinking pretty quickly; in the 3-door scenario, switching from your original choice gives you double the probability of winning, but with 4 doors the advantage is only 50% (.375/.25). With five doors, the winning probability of not switching is .2 while the winning probability of choosing any other unopened door is 4/15 = .2666, for a 33% advantage. With 6 doors, you have a 16.6% chance of guessing correctly on the first try and a 20.8% chance of winning if you switch, for a 25% advantage. We can project from this that the advantage of switching versus staying is $\dfrac{1}{N-2}$ for any $N>2$.

share|improve this answer

Here is an example a friend of mine once told me that I think really makes the monty hall problem really clear. Suppose there are one million doors, one with a golden Bugatti and the rest with an assortment of goats from all parts of the world.

At first you chose a random door. Then Monty opens all the other doors except for the one you have and one other. Then even your intuition tells you you should change the door.(this is different from the one you asked, but it's a great border-case example that is really intuitive).

share|improve this answer
    
Following the reasoning behind the accepted answer, this happens to be a decent (albeit extreme) example. If you open all doors except for the one that has the car, you get (n-1)/n * 1 > 1/n. At n = 1 million, the left member is clearly much bigger than the right one. –  Doktoro Reichard Dec 16 '13 at 17:59
1  
Note that this is only true if Monty intentionally chooses doors with goats. If you pick an arbitrary door, and Monty randomly opens 999,998 doors, and all of them happen to be goats, the probability of the car being behind each door is now 1/2. –  Tanner Swett Dec 16 '13 at 19:13
5  
@Tanner:Then that wouldn't be the Monty Hall problem any longer. –  Dunk Dec 16 '13 at 22:18
    
@dunk Tanner is simply emphasising the importance of the host's knowledge. It's a crucial assumption in the formulation of the problem and should be made explicit when stating it. –  Daniel Rust Dec 17 '13 at 13:52

Increasing the number of doors is a common extension of the problem, and is analytically useful for giving people an intuition about why switching is optimal. This is not, strictly speaking, the situation you're describing (since your question is about adding a single door and Monty still only opening one door), but it's a similar case.

Consider a Monty Hall Problem of 100 doors: Let's say you choose one, then Monty opens 98 other doors. Which probability is higher: the probability that you picked the right one from the beginning (1%), or the probability that you didn't and thus the one door that Monty left unopened is the real winner (99%)?

share|improve this answer

You should switch. By picking one of the doors first, the probability of getting a car is 1/4. Once Monty opens one door with the goat, the probability that the car is in one of the other 2 remaining doors is 1/2 * 3/4 = 3/8 > 1/4. So you should switch. The wikipedia article http://en.wikipedia.org/wiki/Monty_Hall_problem is a good read by the way.

share|improve this answer
    
There are two remaining doors after Monty has opened one door. The 1/2 represents the 1 out of 2 doors remaining, and the probability 3/4 is due to the remaining probability (1 - 1/4) after you have chosen the first door and before Monty opens one door with the goat. –  ireallydonknow Dec 16 '13 at 11:46

protected by Ragib Zaman Dec 18 '13 at 11:32

Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.