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I was watching the movie $21$ yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability.

The question goes a little something like this (I'm paraphrasing, but the numbers are all exact):

You're on a game show, and you're given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $1/3$ chance of winning. Which door would you pick?

The character picks A, as the odds are all equally in his favor.

The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such.

The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $1/3$ chance of winning if door A is picked and $2/3$ if door B is picked.

What I don't understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two?

I assume that I'm wrong; as I'd really like to think that they wouldn't make a movie that's so mathematically incorrect, but I just can't seem to understand why this is the case.

So, if anyone could tell me whether I'm right; or if not explain why, I would be extremely grateful.

amWhy
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Avicinnian
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    This is known as the [Monty Hall problem](http://en.wikipedia.org/wiki/Monty_Hall_problem). The point is that your odds of winning with the original door *have not changed*. Since the total odds have to add up to 1, the odds of $B$ being the correct door are now 2/3. In fact, "switching to B" is equivalent to "pick the best of whatever is behind doors B and C" (you know that what is behind B is no worse than what has been revealed behind C), which clearly gives you a 2/3rds odds of winning. The precise conditions of the game are **very** important, though. – Arturo Magidin Jan 06 '12 at 03:27
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    I would expect that this has been asked before; I found two questions asking about variants ([here](http://math.stackexchange.com/questions/65344/a-variant-of-the-monty-hall-problem) and [here](http://math.stackexchange.com/questions/41807/variation-on-the-monty-hall-problem)), but not the plain question. I'm probably just not finding it? – Arturo Magidin Jan 06 '12 at 03:33
  • @ArturoMagidin I looked as well and was shocked not to find one. – Alex Becker Jan 06 '12 at 03:37
  • The best way to verify the Monty Hall solution is to play the game yourself. Find someone else to be Monty and make sure your choices are random! – Chris Burt-Brown Jan 06 '12 at 09:34
  • **Aside:** in the movie, the professor explains to a student who got the incorrect answer that "You forgot to account for variable change." Now, "variable change" is a phrase that I didn't hear once in eight years of studying mathematics. Is it more common in the US, or did they just make it up for the movie? – Chris Taylor Jan 06 '12 at 10:09
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    I'd suggest that the question is ambiguous as commonly stated. It doesn't specify whether the teacher selected a door at random which just happened to have a donkey behind it, or if the teacher deliberately selected a door with a donkey behind it. – Winston Ewert Feb 06 '12 at 03:43
  • http://math.ucsd.edu/~crypto/Monty/monty.html to play the game with either (1) the usual variant or (2) host picks random door. –  May 04 '12 at 06:33
  • Here is a very good paper on Monty Hall problem : http://www.jstor.org/stable/2684453 There is really quite a lot of subtleties. – faceclean May 22 '12 at 21:39
  • however, because this problem gained that much popularity, I will guess they will only give you that option when you chose the car in the first place, thus, you should not change – ajax333221 Jun 09 '12 at 00:57
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    @WinstonEwert It's also necessary to specify that if you initially choose the winning door, the host chooses which remaining door to open uniformly at random. – augurar Aug 09 '14 at 07:34
  • @augurar, how does that matter? Since neither of the other doors are winner, isn't the same regardless of which door is opened? – Winston Ewert Aug 09 '14 at 16:32
  • @WinstonEwert Suppose the doors are numbered 1, 2, and 3, and suppose the host always opens the highest numbered available door. Then if you chose door 1 and the host opens door 2, there is 100% chance that door 3 is the winner (otherwise the host would have opened it instead). But if you choose door 1 and the host opens door 3, the chance that door 2 wins is only 50%. – augurar Aug 09 '14 at 18:43
  • @augurar, clever! If you know something about the how the host will pick the door, you can glean information from his choice. Thanks for pointing that out to me. – Winston Ewert Aug 10 '14 at 01:15
  • Ah probability! You can easily understand this by assuming there are a 100 doors instead of three. You choose a door, then the person opens all the other doors, except the one you chose and another one. Do you switch now or not ? – kaushalpranav Sep 18 '18 at 06:14
  • @kaushalpranav Doesn't your comment merely duplicate https://math.stackexchange.com/a/96833? – NNOX Apps Aug 23 '21 at 07:02
  • Cf. https://matheducators.stackexchange.com/a/15098, https://stats.stackexchange.com/a/1073. – NNOX Apps Aug 23 '21 at 07:06

12 Answers12

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This problem, known as the Monty Hall problem, is famous for being so bizarre and counter-intuitive. It is in fact best to switch doors, and this is not hard to prove either. In my opinion, the reason it seems so bizarre the first time one (including me) encounters it is that humans are simply bad at thinking about probability. What follows is essentially how I have justified switching doors to myself over the years.

At the start of the game, you are asked to pick a single door. There is a $1/3$ chance that you have picked correctly, and a $2/3$ chance that you are wrong. This does not change when one of the two doors you did not pick is opened. The second time is that you are choosing between whether your first guess was right (which has probability $1/3$) or wrong (probability $2/3$). Clearly it is more likely that your first guess was wrong, so you switch doors.

This didn't sit well with me when I first heard it. To me, it seemed that the situation of picking between two doors has a certain kind of symmetry-things are either behind one door or the other, with equal probability. Since this is not the case here, I was led to ask where the asymmetry comes from? What causes one door to be more likely to hold the prize than the other? The key is that the host knows which door has the prize, and opens a door that he knows does not have the prize behind it.

To clarify this, say you choose door $A$, and are then asked to choose between doors $A$ and $B$ (no doors have been opened yet). There is no advantage to switching in this situation. Say you are asked to choose between $A$ and $C$; again, there is no advantage in switching. However, what if you are asked to choose between a) the prize behind door $A$ and b) the better of the two prizes behind door $B$ and $C$. Clearly, in this case it is in your advantage to switch. But this is exactly the same problem as the one you've been confronted with! Why? Precisely because the host always opens (hence gets rid of) the door that you did not pick which has the worse prize behind it. This is what I mean when I say that the asymmetry in the situation comes from the knowledge of the host.

Alex Becker
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    Fantastic explanation! Thanks for the clarification :). – Avicinnian Jan 06 '12 at 04:11
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    +1 `what if you are asked to choose between a) the prize behind door A and b) the better of the two prizes behind door B and C` really cements the idea. – dj18 May 02 '12 at 17:34
  • Good explanation. But I would say that the fact that the host knows the door which not to pick, is more a kind of asymmetry than symmetry. – Cris Stringfellow Dec 20 '12 at 00:39
  • @CrisStringfellow I wrote asymmetry. – Alex Becker Dec 20 '12 at 18:40
  • There's one point that I'd like to make: Before the host opens one of the "bad" doors, each door has an equal probability of $1/3$ of being the "good" door. However, after the host has revealed one of the two bad doors, there are now two doors---one good door and one bad door---left. So now the probability---which is in fact the conditional probability---of either of the other two doors being the good door is $1/2$ because the contestant at this choice has an open choice of either to stick with his original selection or choose the third remaining door. What is the flaw in this reasoning? – Saaqib Mahmood Aug 09 '14 at 03:31
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    @SaaqibMahmuud The fact that there are two doors left does not imply that each one has equal probability of being the winning door. – augurar Aug 09 '14 at 07:30
  • I'm not saying I'm right because I'm probably not, but I take issue with the statement: "This does not change when one of the two doors you did not pick is opened." So suppose we play a different game where there are only 2 doors, with a prize behind one. You have 50/50 chance in the beginning. Now suppose the host opens the door you didn't pick and it's not the prize. To my mind of course the probability has changed. The probability is now 100% that the door you chose has the prize. So given that the probability changes in the 2-door case why would it be any different for the 3-door case? – User Oct 31 '15 at 02:50
  • @AlexBecker you cannot state that "the host always opens the door that you did not pick which has the worse prize behind it" this information is not present in the question nor in the movie. I agree with the answer of Robbie, the movie is stating the Monty Hall paradox in the wrong way. In particular, if you watch the movie the teacher says "the host **decides** to open another door". – Emanuele Paolini Jan 14 '17 at 18:48
  • This response correctly answers a different problem, not the one presented above, which is equivalent to the original vos Savant column. See pp.151--153 here: https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.33.3.144. The current #2 response is good, also see this response: https://math.stackexchange.com/a/3360686 – Joshua B. Miller Sep 18 '19 at 07:24
  • I will clarify: this solution gives the probability of winning if you always switch. That is a correct statement, but answers a different problem. Just as in the famous version of the problem presented by vos Savant, in this version a specific door is revealed (C). The solution here doesn't explain why you are strictly better off switching conditional on this information. An easy test: if teacher opens C whenever she can (otherwise she opens B). In this case you are not strictly better off, even though the method here correctly says that if you commit to switching, you win 2/3 of the time. – Joshua B. Miller Sep 19 '19 at 07:01
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    @JoshuaB.Miller You are nitpicking over whether the specific door numbers are illustrative examples or prescriptive. You have to realize that this is a pop culture math problem and as such it won't be phrased as precisely as you might like. If you return to the original Vos Savant text you will realize your error. She says "You pick a door, say No. 1" not "You choose door No. 1". The intent is clear: the door number is an illustrative example. – Codure Sep 19 '19 at 15:29
  • @Codure While my point is subtle, it is not in error. The problem in this thread is a conditional probability problem. There is no ambiguity/precision issue--the teacher has opened door C. This argument may help people shake away their initial intuition but it does not explain why you should switch conditional on the door having been opened. Instead it explains why you win 2/3 of the time if you commit to switching, which is also true in the easy test example I gave, where the conditional probability is not 2/3. This proves that this argument does not provide the conditional probability asked – Joshua B. Miller Sep 20 '19 at 08:47
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    @JoshuaB.Miller In your "easy test example" you have successfully engineered a scenario where the conditional probability of winning on switch is not 2/3, though I'm not sure how relevant it is given the constraint you've imposed on the teacher's behavior. Regardless, you have ignored my point. You misread the original Vos Savant text and you have erroneously equated that problem to the one above, which is a pretty rough paraphrasing from a movie – Codure Nov 01 '19 at 21:17
  • @Codure Your point that the door is an illustrative example does not negate the fact that a door has been opened in the example. There is still a missing step in the argument explaining why the probability of winning by committing to switch (presented here) is equal to the probability of winning conditional on the door having been opened (whichever door this is). To do that you need to appeal to symmetry and the law of total probability to show that this requested posterior must be equal to the prior that has been shown to be 2/3. – Joshua B. Miller Nov 04 '19 at 03:53
  • @User: the reason why there is a difference between 2 doors and 3 doors, is because with 2 doors there is no longer any unknown to be calculated. That is also why we say the probability hasn’t changed, because we are talking about the decision made BEFORE the reveal, and since the reveal is never the prize it doesn’t introduce any new information about the choice you made. Over the course of 9 rounds, in 3 of them you will have picked the prize the first time. If you always switch then in 6 of them you switched to the prize. – jmoreno Apr 29 '20 at 00:04
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To understand why your odds increase by changing door, let us take an extreme example first. Say there are $10000$ doors. Behind one of them is a car and behind the rest are donkeys. Now, the odds of choosing a car is $1\over10000$ and the odds of choosing a donkey are $9999\over10000$. Say you pick a random door, which we call X for now. According to the rules of the game, the game show host now opens all the doors except for two, one of which contains the car. You now have the option to switch. Since the probability for not choosing the car initially was $9999\over10000$ it is very likely you didn't choose the car. So assuming now that door X is a goat and you switch you get the car. This means that as long as you pick the goat on your first try you will always get the car.

If we return to the original problem where there are only 3 doors, we see that the exact same logic applies. The probability that you choose a goat on your first try is $2\over3$, while choosing a car is $1\over3$. If you choose a goat on your first try and switch you will get a car and if you choose the car on your first try and switch, you will get a goat. Thus, the probability that you will get a car, if you switch is $2\over3$ (which is more than the initial $1\over3$).

Maha
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E.O.
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Here is an equivalent formulation of the Monty Hall problem, which folks on twitter seem to like.

If you solve this problem, you have solved the original. To see the connection, you can imagine boxers hidden behind the doors.

There are three boxers. Two of the boxers are evenly matched (i.e. 50-50, no draws!); the other boxer will beat either them, always.

You blindly guess that Boxer A is the best and let the other two fight.

Boxer B beats Boxer C.

Do you want to stick with Boxer A in a match-up with Boxer B, or do you want to switch?

Intuitively witnessing Boxer B beating Boxer C gives you information about Boxer B. The advantage of switching no longer feels "paradoxical."

Details: If Boxer B is best, then the probability of witnessing Boxer B beat Boxer C is equal to 1. By contrast, if the initially chosen Boxer A is best, then this probability is equal to 1/2 (B&C are evenly matched with no draws). This means that the event you witnessed is twice as likely if Boxer B were best than if Boxer A were best (i.e. 1 vs. 1/2). This is the information you have received. Bayes rule now tells us that the initial 1-to-1 odds in favor of Boxer B in a match-up against Boxer A must double, i.e. there are now 2-to-1 odds in favor of Boxer B being the best boxer. This means there are now 2 chances in favor of Boxer B for every 1 chance in favor of Boxer A, i.e. 2 chances out of 3 total chances. This is equivalent to saying that the probability is now 2/3 that Boxer B is the best boxer. You should switch.

Notes:

This formulation has the advantage of providing clear intuition for the solution to the problem as presented, which is equivalent to the famous version originally presented by Marilyn vos Savant.

By contrast, the current top response since 2012 on math.stackexchange correctly solves a different problem, in particular the prior probability of winning if you commit to switching regardless of which door Monty opens. What that response is missing is an argument connecting the posterior (conditional) probability to the prior. For a justification, see my comment there, or you can read this paper, page 151--153.

Update: I googled and it appears that the first to point out a framing like this were Burns & Wieth, Journal of Experimental Psychology--General 2004.

Joshua B. Miller
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It's simple: switching allows you to pick 2 out of the 3 doors. Choosing door number 1 and then always switching, is the equivalent of saying "door number 2 or door number 3, but NOT door number 1". When you look at it that way, you should see that you have a 2/3 chance of being right, and that the opening of a door simply confirms which door it must be if you are right.

Increase the number of doors and it should become even more obvious that saying "door 2 or 3 or 4 or 5 or ..., but not 1" is the right way to bet. You have an $1-1/x$ chance of being right, and a $1/x$ chance of being wrong.

NNOX Apps
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jmoreno
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  • Doesn't this answer merely repeat https://math.stackexchange.com/a/96833 above? – NNOX Apps Aug 23 '21 at 06:59
  • @ugro: all answers are describing the same thing, with different emphasis and in different ways. I don’t believe this and that answer are similar at all. – jmoreno Aug 30 '21 at 23:05
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The person who changes his choice will win if and only if his first choice was wrong, and there is a probability of $\frac{2}{3}$ on that.

The person who does not change his choice will win if and only if his first choice was right. There is a probability of $\frac{1}{3}$ on that.

drhab
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Let us use some theory here. Lets call the doors $0, 1, 2,$ and the right door $D$. $$P(D=0)=P(D=1)=P(D=2)=\frac13$$ $D$ is random. Now let us call the door we choose $C$. $C$ is not random. Without loss of generality, let $C=0$. Also we have $R$, the revealed door, which is random. Since the person won't reveal the right door or the one you choose, $R \neq C$ and $R \neq D$. Since we know $C$, without knowledge of $D$ we have: $$P(R=1)=P(R=2)=\frac12$$ Let us say, without loss of generality, we get the information $R=2$. Then what we are looking for is $P(D=1|R=2)$. $$\frac{P(D=1 \wedge R=2)}{P(R=2)}$$ $$\frac{P(R=2|D=1)P(D=1)}{\frac12}$$ Now, $P(R=2|D=1)=1$, since $R$ can't be 0 or 1, since those are already taken by $C$ and $D$. $$\frac{1 \cdot \frac13}{\frac12}$$ So the answer is: $\frac23=66.\overline6 \%$.

PyRulez
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The movie 21 didn't state the riddle correctly. The movie failed to state the rules governing how the game show host will behave.

Assuming the riddle in the movie follows the "Monty Hall Problem" as described on Wikipedia there a few critical assumptions the movie failed to mention:

1) the host must always open a door that was not picked by the contestant 2) the host must always open a door to reveal a goat and never the car 3) the host must always offer the chance to switch between the originally chosen door and the remaining closed door.

Knowings the rules it makes the riddle much easier to understand. Many of the explanations above will suffice and Wikipedia has a good explanation.

The problem is that the movie failed to state these critical assumptions.

Robbie
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    It's also necessary that if the contestant initially chooses the winning door, the host chooses either of the remaining doors with equal probability. See my comment above. – augurar Aug 09 '14 at 18:46
  • +1 If you watch the movie you should notice that the teacher says: "...the host decides to open another door..." which means that you cannot know that the door is always opened. – Emanuele Paolini Jan 14 '17 at 18:50
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    Yea, the movie gets this crucial description wrong. Kevin Spacey asks why the student doesn't worry that the host is playing a trick on him ("reverse psychology") since the host knows where the cars is. Indeed, if the host is allowed to choose to open a door *conditional* on whether the player initially picked the right door (which violates the key assumptions augurar identifies) then the player can be wrong. For instance if the host reveals a door only when the player initially chooses correctly, and the player always switches when offered the chance, then the player never wins. – Jess Riedel May 14 '17 at 17:08
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Merely knowing that the teacher showed a losing door does not provide any information unless one knows how the correctness of one's initial answer would influence the likelihood of the teacher showing the losing door. Consider the following four possible "strategies" for the teacher:

  1. The host knows where the prize is, wants the contestant to lose, and will show an empty door only if the contestant had picked the one with the prize [if the contest already picked a wrong door, the host would reveal either the contestant's door or the one with the prize].

  2. The host knows where the prize is, wants the contestant to win, and will show an empty door only if the contestant had picked the other empty door [if the contestant had already picked the right door, the host would simply show it].

  3. The host knows where the prize is, and will always show an empty door [the empty door if the contestant's initial guess was wrong, or an arbitrarily-selected empty door if it was right].

  4. The host picks a door at random; if it contains the prize, the contestant loses; otherwise, the contestant is allowed to switch to the other unseen door.

In the first two scenarios, the host's decision to show or not show an empty door will indicate to anyone who knows the host's strategy whether the player's guess was right or not. In the third scenario, the host's decision to offer a switch provides no information about whether the contestant's initial guess was right, but converts the 2/3 probability that the contestant's initial guess was wrong into a 2/3 probability that the prize is under the remaining door.

To evaluate the last scenario intuitively, imagine that the host draws an "X" on the player's door, flips a coin to pick a door at random and draws an "Y" on it, and finally draws a "Z" on the remaining door. If neither the host nor player has any clue as to where the prize is, doors 1, 2, and 3 will each have an equal probability of holding the prize, and the marking of letters X, Y, or Z by people who have no idea where the prize is don't change that. If the host asks the player if he'd like to switch to Z before anyone knows what's under Y, the decision will be helpful 1/3 of the time, harmful 1/3 of the time, and irrelevant 1/3 of the time. If door Y is shown to be empty, the irrelevant case will be eliminated so of the cases that remain, the other two will have 1/2 probability each.

Note: Many discussions of the "Monty Hall Paradox" assume that the host uses strategy #3, but fail to explicitly state that fact. That assumption is critically important to assessing the probability that a switch will be a winning move, since without it (depending upon the host's strategy) the probability of the prize being under the remaining door could be anything from 0% to 100%. I don't know the strategy used by the real-life game-show host for whom the "paradox" is named, but am pretty certain I've seen players revealed as winners or losers without being given a chance to switch, implying that while Monty Hall might sometimes have used strategy #3, he did not do so consistently [the normal arguments/proofs would hold if the host's decision of whether or not the player would be shown an empty door and allowed to switch was made before the player selected his door, but I have no particular reason to believe Monty Hall did things that way].

supercat
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Suppose you and your brother play simultaneously .You and he always choose the SAME door to start. You NEVER switch,so after your initial choice you go out for a coffee and come back after all 3 doors are open. Your chance of winning is unaffected by what happens in between.It's 1/3. Your brother ALWAYS switches. EVERY TIME YOU LOSE,HE WINS. YOU LOSE 2/3 OF THE TIME.

DanielWainfleet
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  • Doesn't this merely recapitulate https://math.stackexchange.com/a/96832 and https://math.stackexchange.com/a/96833 above? – NNOX Apps Aug 23 '21 at 07:00
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Rather than looking at the player, I prefer to explain the paradox from the host's standpoint, as this only involves one step.

As the player gets one door, the host gets two. There are 3 possibilities with the same probability:

  • donkey-donkey => leaves a donkey after a door is open
  • car-donkey => leaves the car
  • donkey-car => leaves the car

So in two cases out of three the door that the host leaves closed hides the car.

Christophe
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The tip I've come up with to explain this problem is that you need to consider that the doors AREN'T RANDOMIZED after Monty deletes the door that is wrong. This means that although the probabilities the two remaining choices have of being correct add up to 1, the two probabilities aren't the same as before. It is wrong to state that because they had equal probabilities before, 1/3 each, they should have equal probabilities after Monty's deletion. The doors are randomized in the beginning when you have three doors, which is why you can say that each option has and equal chance of being right. However, after Monty deletes the wrong door from the other doors which you did not pick, now you are left with the probability that the door you picked is right of 1/3, and the probability that the other door is right, which is also equal to the probability that you chose the wrong door of (1/3 + 1/3 = 2/3). Well, those are my 2 cents, and I hope it helps!

Ryo
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  • Doesn't this merely recapitulate https://math.stackexchange.com/a/96832 and https://math.stackexchange.com/a/96833 above? – NNOX Apps Aug 23 '21 at 07:01
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Let us compare our chances of winning when we switch and when we don't.

Scenario 1-Switching:You switch your choice after the host opens the door not containing the prize.

Now if you want to win the prize, initially choosing the door with no prize and the scenario-1 condition will guarantee your win.And the probability of you initially choosing the door with no prize is 2/3. (Because there are two doors with no prize) .

So the probability of you winning in Scenario 1 is 66.66%.


Scenario 2-Not switching: You don't switch after the host reveals a door with no prize.

Choosing the correct door initially and the scenario-2 condition will guarantee your win. And the probability of you choosing the prize door initially is 1/3.

So the probability of you winning in case of Scenario-2 is 33.33%.


Rohan Marwah
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  • Doesn't this merely recapitulate https://math.stackexchange.com/a/96832 and https://math.stackexchange.com/a/96833 above? – NNOX Apps Aug 23 '21 at 07:01
  • @Velodyne it's about explaining the answer in the most intuitive way. Most of the answers are trying to explain the same solution. – Rohan Marwah Sep 12 '21 at 09:57