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Re: Let us be serious for once

edited 30 November -1 at 12:00AM in The Money Savers Arms
32 replies 1.4K views
6.2K posts
edited 30 November -1 at 12:00AM in The Money Savers Arms
Robert Shilling Presents   ...

The Monty Hall Dilemma   ...

It is fairly easy to understand the question  ...

You have won a TV Quiz show.

You can open any one of three doors.

Your prize will be the amount of money behind the door you open.

Behind one door there is £10,000

Behind each of the other two doors there is  a  5 pence coin ( i. e.  A Shilling in old money. )

You know this

The doors are marked as A, B and C.

You do not know what is behind any particular door  but the quiz master knows exactly what is behind each door.

You are invited to say which door you choose.  A,B or C.

After you have told the quizmaster your choice he opens one of the remaining two doors and shows you that it contains a Shilling coin.

He then asks the contestant if she would like to stick with her original choice or change to the other remaing door.

What should she do.

1) Stick to her original choice?
2) Change to the other remaining door?
3) It makes no difference?

What is your answer.

1, 2 or 3

This question has baffled intelligent people throughout the world including professors of mathematics.

Rob - should this not be posted in the money saving arms ?


  • Robert_Sterling_3Robert_Sterling_3 Forumite
    7.1K posts
    You are quite right of course

    If you remove your post then I can remove this post and the thread I think

    Or perhaps a passing moderator could delete the thread.
    ...............................I have put my clock back....... Kcolc ym
  • The answer is pretty simple!

    Ask the quizzmaster to close the two open doors and open the remaining one!

    NB the quizzmaster has cheated on the rule, because it is the prize behind YOUR choosen door and NOT that of the quizzmaster.

  • First off, apologies for replying to this thread, when, as Deemy points out, it really should be in the Money Savings Arms, but it is one of my favourite statistical conundrums, and I haven't seen it reposted over at the MSA. Don't bother reading this posting if you want useful credit card information.

    Next, to reply to Systemaddict: I think you've misunderstood. At the point where the contestant has the chance to switch or stick, the door she has chosen is still shut, so there is still a chance that she has chosen the correct door. But the real question is, what is that chance, is it 1 in 3, 1 in 2 or something else?

    And now my thoughts on the original question:
    Now the usual fallacy is to say, that there are still two doors left, one has 5p and one has £10,000, so each door has a 1 in 2 chance, so the answer to Mr Sterling's original question is '3) It makes no difference'.

    However, if we make an assumption about the quizmaster, then we can show that in fact the other unopened door, that the contestant hasn't chosen, has a 2 in 3 chance of having £10,000 behind it, so the correct answer to Mr Sterling's original question is '2) Change to the other remaining door'.

    The assumption that we have to make is that whether or not the contestant initially chooses the correct door, the quizmaster always opens an unchosen incorrect door.

    It might not be true that the quizmaster is consistent. If the quizmaster was a very nice person, he might only open the unchosen incorrect door if the contestant has first chosen an incorrect door. Alternatively, if the quizmaster was a not very nice person, he might only open an unchosen incorrect door if the has first chosen the correct door!

    However, if we allow these possibilities then we are in to the realms or physchology rather than statistics, so instead we assume that the quizmaster is unbiased and always opens an unchosen incorrect door.

    When the contestant initially chooses one of the three doors, anyone who understands probabilities or odds will agree that she has a 1 in 3 chance of being correct. The harder question is what happens when the quizmaster opens an unchosen incorrect door. This gives us some new information, and so changes the odds of the remaining doors, but how?

    Another fallacy is to say that the 1 in 3 chance that has now been removed when the quizmaster opens an unchosen incorrect door is shared out equally between the two remaining doors, which leaves them each with a 1 in 2 chance. This would be true if the quizmaster didn't know what was behind each door, and it was just chance that he chose a wrong door, but the question states that 'the quiz master knows exactly what is behind each door'. Now if the contestant has already chosen the correct door, then the quizmaster can just randomly choose one of the two unchosen incorrect doors. But if the contestant has chosen incorrectly, then the quizmaster's hand is forced, and he has to use his knowledge to chose the only unchosen incorrect door.

    One third of the time the contestant chooses the correct door initially, and then switching is a bad move.
    But two thirds of the time the contestant chooses an incorrect door initially, and then switching is definitely the correct thing to do, as the other remaining door, after the quizmaster has ruled one out, must be correct.

    I remember when I was first introduced to this, and I was convinced that the answer was 1 in 2, but after much thinking I believe I understand it correctly now, and thinking about the underlying assumptions helped also.

    Another useful example that can help to consider is what if there were 1,000,000 doors, the contestant chooses one, and then the quizmaster open 999,998 incorrect ones. In this case I would say that the contestant would be mad not to switch, but of course this is based on the same assumption about the quizmaster.

    Another thought is back to psychology again rather than statistics. You might feel, even knowing the odds, that you would be gutted to switch from a 1 in 3 chance to a 2 in 3 chance and then find that you were right in the first place. You might decide, if you felt that way, to stick, to avoid the bad feeling in that 1 in 3 case, but the correct financial decision would be to switch.

    I guess this does have some relevance to money saving, as many financial decision, especially insurance related, do have an element of understanding risks or probabilities.

    Sorry to ramble on so long!,

  • Robert_Sterling_3Robert_Sterling_3 Forumite
    7.1K posts
    Would anyone else like to contribute.

    DD has opened the bidding with a vote for changing to the remaining unopened box and thinks it is twice as likely to have the £10,000 cheque in it as the original box is.

    When this Dilemma first surfaced there were professors of mathematics who supported DD's explanation and there were professors of mathematics who said that there were two boxes left and the £10,000 was equally likely to be in either one of the two remaining boxes.
    ...............................I have put my clock back....... Kcolc ym
  • Im not sure about this but Ill tell you what I think.

    At the start the player has a 1/3 chance of finding the £10k and he picks a door (i.e. 1) Once he has picked this door one is revealed not to have the £10 (it cant be the dorr picked)

    Now, the odds have cut to 1/2 for finding the 10k but it makes no difference which door is picked out of the remaining two.

    Therefore, sticking with the same door or changing will not affect the outcome. Although changing and finding out that you changed from the £10k doo will hurt alot!
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  • nhnh Forumite
    567 posts
    I think it is equally likely to be behind either of the other two doors, though I understand DT's explanation.

    This is very interesting by the way and has kept me occupied for ages (in between actually doing some work).

    DT says that having already chosen one door, it is more likely that the door the contestant didn't choose is the right one.

    But, if the contestant came to the doors after one had already been opened, he would stand a 1 in two chance of choosing the right door.

    The quizmaster opening a door is a red herring. The contestant can choose any door and it doesn't matter, the quizmaster will still be able to take one of the wrong doors out of the equation. Whether he has two of these wrong doors to choose from or not is insignificant (in my opinion ;)). The fact that he will always have one he can open is what is important.

    So after the quizmaster has opened one door it goes back to a straight choice between one right door and one wrong door, irrespective of whether you chose the right or wrong door in the first place.

    Therefore you have a 1 in two chance of getting the right door if you stick.

    The same applies even if you have 1,000,000 doors!
    I'm married now! Yippee!
  • Robert_Sterling_3Robert_Sterling_3 Forumite
    7.1K posts
    Just to add to the fun .....
    Suppose another person appears on stage.
    She knows nothing of what has gone on so far.
    She is told, quite truthfully, that there are two unopened boxes on stage. One has £100 in it and the other has £10,000 in it. What are her chances of picking the right box if given a choice.
    ...............................I have put my clock back....... Kcolc ym
  • david78david78 Forumite
    1.7K posts
    Switch. If you switch you have twice as much chance of winning than if you don't. Switch chance is 2/3. Don't switch chance is 1/3.


    DT is spot on.

    The "newcomer on stage" has a 50-50 chance of winning. If he/she saw the box picked by the original contestant, then he/she should of course pick the other one.
  • The second contestant is picking boxes, not doors
  • david78david78 Forumite
    1.7K posts
    Here's an explanation of why you should switch.

    Your first guess had only a 1/3 chance of being correct.

    Now suppose the quizmaster says you can swap that one door and open the other two doors instead. You would, wouldn't you. (Go on you know you would - it gives you a 2/3 chance of winning).

    Well that's effectively what he does say except he saves you the trouble of opening one of the other two doors by pointing out that it contains 5 pence.

    So you should switch.
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