The Monty Hall Problem: Still Hanging Around

I can’t believe that this problem is still hanging around. The latest version is over at Dean’s World and reading the comments is…well painful. So, here is a formal proof that switching is always better than sticking.

First, there are three doors, Door 1, Door 2, and Door 3. Now for some notation:

Ci: Denotes the event that the care is behind door i.
Onot i: Denotes the event that Door not i is opened by Monty.

These are the relevant probabilities:

  • P(Ci) = 1/3; this is the probability that the car is behind door i, where i = 1,2,3.
  • P(Onot i) = 1/2; this is the probability that Monty opens door not i given that the care is behind door i.
  • P(Onot i|Ci); this is the probability that Monty Opens door not i given that the car is behind door i.

Now what is the probability that the car is behind door i given that we are playing the strategy “stick” (i.e., we stick with our orignal choice) and that Monty has opened one of the doors we did not pick. That is,

P(Ci|Onot i) .

By Bayes Theorem we can write,

P(Ci|Onot i) = [P(Onot i|Ci) * P(Ci) ]/P(Onot i) .

Now, on the right hand side of the equation the first term in the numerator is 1/2 since the door is behind Ci Monty can pick either of the two doors not picked randomly. The probability in the numerator is also 1/2, and the “prior probability”–i.e. P(Ci)– is 1/3. Thus, the 1/2’s cancel out and we get a probability of 1/3. That is the Probability that the car is behind door i is 1/3 if we stick.

Now suppose we play “switch”.

We again use Bayes Theorem to come up with,

P(Ci|Onot i) = [P(Onot i|Ci) * P(Ci) ]/P(Onot i) .

Now we have chosen the door Monty did not open. Now, given that the car is behind the door we switched too, prior to that switch Monty’s only choice of door to open was the other door, hence the probability for opening not i is 1. The probability in the denominator is still 1/2, and the prior probability is 2/3.

Q.E.D.

Note, that in this proof I have not made any assumptions about which door was chosen by the player. Note also, that this is a mathematical proof. In other words, the switch strategy has been proven (i.e. 100% beyond doubt) to be the superior strategy. You can disagree if you want, but unless you can find a problem with this proof your arguments are wrong. Feel free to post any reason you think switching is no better than sticking, but unless you can show a flaw in the proof you are wrong.

Update: I thought I’d add some explanation at to why switching is the better strategy. When you first pick a door, you know nothing about each door. So when you pick chances are you picked wrong; after all your chance of getting the right door is 1/3 and the chance the other two doors have the car behind one of them is 2/3. Now, when Monty opens one of the doors you did not pick Monty is telling you something about the doors you did not pick, not about the door you did pick. In other words, Monty is telling you about the set of doors that is most likely to win, and switching takes advantage of that information while sticking ignores that new information.

If you are still not convinced consider a variant of the game. Suppose there are 1 million doors. Is your first choice likely wrong? The answer should be yes. Now after your first choice Monty opens up 999,998 doors that don’t have the car behind them that you didn’t pick. Now why is the door you first picked suddenly the likely winner? It is still has a probability of 1/1,000,000 of being the winning door. So the door that is still closed and you didn’t pick is probably the winner (with a probability of 99.9999%).

If you still aren’t convinced, then please…please, never go to Las Vegas or Atlantic city for anything other than the shows.

Update II: Damn it, but Rodney Dill, OTB contributor, has come up with a very elegant way of showing why switching is always superior. When Monty opens one of the two doors you didn’t pick and offers to let you switch to the un-opened door you are basically being allowed to pick both of the doors you didn’t pick the first time. Hence the probability of switching is indeed 2/3. Thanks Rodney.

FILED UNDER: General, Science & Technology, , ,
Steve Verdon
About Steve Verdon
Steve has a B.A. in Economics from the University of California, Los Angeles and attended graduate school at The George Washington University, leaving school shortly before staring work on his dissertation when his first child was born. He works in the energy industry and prior to that worked at the Bureau of Labor Statistics in the Division of Price Index and Number Research. He joined the staff at OTB in November 2004.

Comments

  1. James Joyner says:

    reading the comments is…well painful

    This is pretty painful, too, if you’re not a math geek!

  2. Steve Verdon says:

    I know, but any type of non-rigorous explanation simply does not work, there are those who will insist that the probabilities become 50-50.

  3. David C says:

    I think the key thing causing misunderstandings of this problem is people not getting the key fact:

    Monty KNOWS where the car is!

  4. James Joyner says:

    Heh.

    The problem with your update is the same as those in Dean’s comments section.

    Intuitively, what you have after eliminating one door as the known zonk is only knowledge that the grand prize is either behind the door you picked initially and the remaining unrevealed door. Ditto with the million doors. The chances that it is your door are 1/million initially but, presumably, each door revealed as “not it” incrementally increases the likelihood that your door are therefore “it.”

    The math and “common sense” just don’t match up in this case.

  5. RJN says:

    I think this worked for me once. I sent the puzzle to a friend 8 months ago or so, and after four or five lengthy efforts to explain I came up with this. I think it worked because we never talked about the puzzle again.

    There are three closed cupboards. One of these cupboards has a prize in it. You don’t know which one, and Monty doesn’t know which one, has the prize in it.

    Monty is going to let you choose two of the cupboards so that you will have a 2 in 3 chance of winning. First, however, you must set aside the one you don’t want. Fine, you have chosen; do you still have a 2 in 3 chance of winning? Yes, of course.

    O. K. As you know one, of the two, cupboards you have chosen to keep must be empty. O. K. do you still have a 2 in 3 chance? “Yes, of course.”

    If I show you which one is empty will you still have a two in three chance? “Yes, of course.” Here, a stagehand will show you which one is empty. Do you still have a 2 in 3 chance? “Yes, of course.”

    This explanation was, in my example, preceded by lengthy discussion. As I say, I think it clicked.

  6. Fred says:

    I never had a problem understanding this. However, watching others struggle with it has made me despair for trial by jury and the democratic process in general.

    Then again, “There is a voting booth behind one of these three doors…”

  7. Herb says:

    Steve:

    I started reading your post prior to observing who wrote it, after reading a few lines, I knew it was you.

    Steve, no one gives a damn about 3 doors or most of the other subjects you write about. It does not effect them one bit.

    You did a lot better and got a lot more responses when you wrote about what effects their pocketbook, like OIL Prices.

    Please tell us, Why oil prices are going down, Is there suddenly less demand for oil” or has the threat of placing the oil company execs under oath effecting the price of oil and gasoline? are we now coming into an “oil glut” period? Talk about stuff like that Steve, you do a lot better than talking about doors 1,2, or3.

  8. Rodney Dill says:

    It was this problem that Dean posted several years ago that got me reading Dean’s World. I think the site was referenced from slashdot.org at that point because of the post. Some post on Dean’s World point to some sort of Caption Contest at Outsidethebeltway. And sometime after that I ‘discovered’ Wizbang. So if James ever gets annoyed at me he can blame Dean.

    And yes I do understand the math.

  9. RJN says:

    Herb: You are a rude bloke. The price of oil is responding to market forces. Duh. The company you work for, probably tax payer supported, as a quess, gets better pay and benefits that oil company workers.

  10. Deskmerc says:

    Try it with this.

    A quick test using the “pick the other door” strategy gives me this after 20 runs:

    Wins: 13
    Losses: 7

    I win 66% of the time. Isn’t that close to 2/3?

    Another run of 20 using the “stay pat” strategy:

    Wins: 4
    Losses: 16

    Only won 20% that time. Sucks.

    If I were to run many, many more trials, the results would then tend to rest near the statistical results provided by the proof. You can’t argue with empirical, unbiased results, even if you don’t understand the math.

  11. Steve Verdon says:

    Herb,

    Actually it does matter. Most people just don’t get this at first (some never ever get it). It highlights that people are not rational when it comes to evlauation of probabilities…this applies to real life not just game shows. Such as insurance policies, investing, and so forth. The game show example, is just a cute way of showing that most people don’t think according to the laws of probability.

    Deskmerc,

    Dang, I should have googled for such website. Nice find.

  12. Rodney Dill says:

    Another way to look at it is you keep the original door, or you get to choose both of the other doors.
    The fact that Monty opens the one that has to be empty is inconsequential.

  13. Steve Verdon says:

    Rodney,

    Dang it! How dare you come up with an elegant way of explaining the problem.

    grumble, grumble, mutter, mutter

  14. James Joyner says:

    Rodney/Steve:

    But don’t you get both doors either way? That is, Monty’s giving you one free door plus either your original door or another. Either way, you know the “new” door doesn’t have the prize.

    I know this works mathematically but this explanation doesn’t help make it more intuitive.

    Amusingly, when I played around at the Java site linked in the comments above, it took me well over forty iterations to finally get behind by sticking to my original every time and I got bored before I fell below 44%. Their explanation is similar to Rodney’s:

    Assume that the dealer does not show you the one losing card. In that case, the dealer is saying to you, “Keep your card or choose the other two cards.” Even though you know it can’t be both of those cards, you know there is a 66 percent chance that you will win by switching.

    That’s of course true.

  15. RJN says:

    I think I gave the “Monty gives you two doors solution” above at comment #5.

  16. Herb says:

    RJN:

    Wow RJN, What is a “bloke”. You must be a liney or an Ausse. If you would have bothered to read my comment, you would know that the so called “market forces” is what I was commenting about. Perhaps what you did read just didn’t sink in and that is ok with me, but next time, reply with a comment that would provide a reasonable answer.

    As for the last part of your comment, again, you don’t know what the hell you are talking about and your guessing leaves a lot to be desired. And for your information, I am retired from many years in the Airline Industry that is not supported by the taxpayer. The tax supported airlines and aircraft manufacturers are common in the EU. Not Here Bloke.

  17. Rodney Dill says:

    RJN, yep it’s the same solution, these are all pretty much different ways of paraphrasing the solution. I just chopped it down to fewer words enough to annoy Steve and have him label it ‘elegant’ woo hoo 😉

    As I said earlier I saw this at Dean’s World in what was probably late 2002 or early 2003. To finally convince myself that solution of switching was correct I had to change my belief to being that switching was correct, instead of trying to prove it wrong. The million door selection analogy also helped me visualize the problem.

  18. RJN says:

    Dill: I tried the simple fewer words several times on my friend without success. Eventually I went to the lengthier explanation, with a confirmation at each step of the way, to cement understanding.

    You really have to try to explain this to someone and then you will understand why the need for extra words.

    Herb: I may be a bad boy; I am not sure yet. I assumed that you were railing at some imaginary malfeasance by officers of private corporations because they like to make a buck for their stockholders.

    Attempting to place oil executives under oath in a committee hearing was a stupid, and insulting publicity stunt by the Dumbocrats. They were only interested in pictures for their MSN toadies.

    Long playing government bungling has led to the sad plight of the Airline Industry. We all regret the upheavals energy cost, and increased safety costs has brought.

    However, the unions are not blameless. They have priced themselves out of the affordable range of the airlines. Union pension and health care costs, are way above the norm for wage earning and non-government salary earning America.

    I thought you were swinging like a bloke, so I called you a bloke. Being a bloke is not bad; being rude may be. If you were not railing, then I was wrong to assume you were.

  19. physics geek says:

    Painful comments? I believe that I included the very same probability proof as you, albeit without the fancy mathematical nomenclature.

    People will argue this one with me until they’re blue in the face, but after empirically testing the hypothesis, they always come around.

  20. floyd says:

    RJN; i must take issue with your statement that the officers of private corporations just want to make a buck for their stockholders. in the case of big oil ,with whom i have a long standing relationship, they have made obscene profits for some time now. almost none of said profit has gone to stockholders, nonexecutive employees, or even capital improvement. there was a time when these officers clearly were motivated as you ascribe;however an unjaundiced eye would question whether that is presently the case.

  21. Steve Verdon says:

    floyd,

    So the stock for these companies has gone down then?

  22. floyd says:

    steve, down in the face of recent escalating margins short term , with miserable dividends. overall profitable long term 30+ years, but not to a degree to make me believe the stockholders are their primary concern of late. i admit that my position is based on knowing the people in the field. i also am no economist and i understand there are many mitigating factors of which i am unaware. a reasonably intelligent person can sometimes know when he’s being hoodwinked without knowing the exact method employed.to quote my old algebra teacher,” you know son, liars may figger, but figgers don’t lie”. because the issue is so complex i am left only with what i consider good reason to question [not define] their motives.

  23. Steve Verdon says:

    Really? My co-worker bought some Valero a while ago when it was cheap, not it is worth something like 6x what he paid for it. So, you’ll have to forgive me if I don’t subsribe to this notion that the stockholders are getting screwed.

  24. floyd says:

    focus; the question was motive.