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Puzzles, Jokes, Anecdotes, and Thoughts

Discussion in 'General' started by pianolady, Dec 14, 2007.

  1. rsmullyan

    rsmullyan New Member Piano Society Artist

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    Reply to Terez: I have heard this lovely promlem before, and I did get it right. I think your idea is excellent that those who have heard problems before should say so, but not post the answers.I hope you all will do that with problems that I post in the future.

    Raymond
     
  2. Terez

    Terez New Member

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    I figured you would have. ;) I'm actually surprised that you didn't come up with it! :lol: I came to know the problem through reading a nice after-the-fact summary of a fiasco that occurred when a certain lady who once claimed the world's highest IQ answered it in her column. And even after it was all nicely broken down, she still had math teachers writing in calling her explanations "female logic". :lol: I didn't figure it out on my own, though...I was just reading about it because Wikipedia had a citation of it on her page, and external links to her webpage where the summary of the fiasco was. It actually took a little bit of effort to wrap my head around the logic of it - the only reason I think I was able to is that she mentioned all of the teachers who thought she was wrong who performed controlled experiments in their classrooms fully intending to prove her wrong...and ended up proving her right. :D

    I don't recall the bookworm one. I just knew that discounting the outer covers just wasn't quite tricky enough to make it riddlish. :D There have been a few on here that I already knew, though. Mum's the word. ;)
     
  3. techneut

    techneut Active Member Piano Society Artist Trusted Member

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    Isn't that a logical contradiction ?? :p gdrrrrrrr............................
     
  4. Terez

    Terez New Member

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    If it exists, it's probably not as contradictory as it might seem. Just something one might not yet have come to understand. 8)
     
  5. rsmullyan

    rsmullyan New Member Piano Society Artist

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    I am sorry Techneut, but you are are wrong in your answer to Terez's problem!If the other door were opened at random you would be right, but the one who opened the door deliberately chose a door he knew contained a goat, and this makes a big difference!

    As for you, Terez, I hope you have an elegant way of convincing people that the correct answer is really correct ! Fortunately I have one.

    This problem has a very interesting history: it was posed in a newspaper by a non-mathematici named Marylin , who gave the correct answer and received a storm of angry letters from professinal mathenaticians-- Ph Ds in math--all over the country who all had the same wrong answer, and who harshly berated Marylin fot her imcompetence! But Marylin was right, and the professionals were all wrong, and so dam self-rightious about it! I don.t know if she ever convinced them of the correct answer, but i have thougt of a way of doing so. And so, Terez, I hope that after you have given your explanation of the correct solution,you won't object if I also give mine!

    Terez, I really admire your intelligence enormously!

    Raymond
     
  6. techneut

    techneut Active Member Piano Society Artist Trusted Member

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    Yeah, I suppose it does. If I were the contestant, I would assume that, by offering to change my choice, (s)he was trying to lure me away from the door which (s)he knows hides the prize, and I'd stick to my choice. But this has nothing to do with maths - and it assumes the quizmaster doesn't WANT me to draw the prize, which may be all wrong. Otherwise, I still do not see how one of the two remaining closed doors could be favoured above the other.
     
  7. Terez

    Terez New Member

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    That's one of the problems with stating puzzles like this....you have to eliminate unwanted variables with your wording of the puzzle, and this is one of those that's usually worded much worse than mine was. You can assume that you, the contestant, are familiar with this show, and you know that the offer to switch doors is given every time this game is played, so there is no motive behind the offer.

    I think I do - I'll wait until tonight to post it, though. Some people like suspense...and the impatient ones can google it. ;)

    Yours will probably be more creative than mine, so why not? Marilyn tried a few different ways to make it clear....probably because she knows that different analogous explanations work better for different people. :D Once the position has been tested and proven, as this one has...the more creativity within the confines of the proof, the better!

    Jeez! I'm not the one that figured it out. :p
     
  8. pianolady

    pianolady Monica Hart, Administrator Staff Member Piano Society Artist Trusted Member

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    Can the people behind the scenes change the prizes around?
     
  9. robert

    robert Active Member Piano Society Artist Trusted Member

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    Ah, I love riddles. Tend to forget to listen to the recordings though ;).
    And I swear I have never heard this before!

    First, we must assume that the host always would open a door with a goat no matter if the player picks the right door from the beginning! Ok?

    If you have picked the right door from the beginning, you would be left with a 50% chance but as there is only 33% to really pick the right from the beginning, it is greater chance that the door you have not picked is the correct. The probability for staying on the door will be (50% + ~33%)/2 ~ 42% The probability when switching would be (50% + ~66%)/2 ~ 58%.

    So, I would definitely switch. :D
     
  10. Terez

    Terez New Member

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    Nope. There are no hidden tricks, here - it's not really a riddle so much as a probability puzzle.

    I will ask a provocative question before I call it a night (it's dawn here, but whatever - I keep odd hours :lol:) - just because some more discussion on this one will be fun:

    When you are first given the choice between the three doors, the probability is 33.33...% that any given door conceals the prize.

    You choose your door; the host reveals the goat behind one of the other two doors.

    So...what, exactly, makes you think that, just because one door is now proven to be 100% certainly not the prize, that the door you already chose has any different odds of being the prize than it did in the first place?
     
  11. pianolady

    pianolady Monica Hart, Administrator Staff Member Piano Society Artist Trusted Member

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    Terez, is Robert right? I haven't had my three cups of coffee yet, so it's too hard to think this early.
     
  12. techneut

    techneut Active Member Piano Society Artist Trusted Member

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    Sure. He would hardly open the door with the prize behind it, would he :lol:

    There seems to be a whole area of science proving that you should switch. But I still stubbornly believe it does not matter. The fact that you had already picked a door seems irrelevant to me, as you are given a new chance in the new situation. Now you have the choice between two doors, one of which has the prize. What can be simpler than that !?

    Just suppose, you had not picked a door the first time, and the host had opened his door directly. Would that influence either the choice of the host, or your final choice ? Can't think why it would. And in this case, everybody would agree to having a 50% chance.
     
  13. techneut

    techneut Active Member Piano Society Artist Trusted Member

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    Bacause :!: that 33.33% chance was when there were three choices. Now you have only two, and are given another choice. Adapting to a changed situation seems to me the smart thing to do :wink:

    Actually there is no such thing as a probablility that the prize is behind one particular door. It either is, or it isn't. The probablility element is only introduced when you have to choose.

    This makes me think of these billboards we have in Holland, from some insurance company, which show statements like

    There is an 78% chance that you sometimes drive too fast.

    This is utter crap. The chance that I sometimes drive too fast is 100%, because I sometimes do. But If I read this and I am not a driver of any vehicle, it is still wrong, as in 0%. To be anywhere near useful, a statement like this should be put in combinatorial terms, like "78 out of 100 randomly chosen drivers sometimes drive too fast". It is always dangerous to put mathematically inclined puzzles or statements in non-mathematical (and therefore imprecise) terms.
     
  14. Terez

    Terez New Member

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    Another way of stating it that would probably be more helpful to solving the problem is that probability is only relevant when there are variables - unknowns.

    Fortunately, mathematics applies to quite a lot of apparently non-mathematical situations. This is one of them. ;)

    Keep in mind that this has been tested. Any number of controlled experiments have shown that there is indeed an advantage in switching doors. This truth is quite fortunate for the game contestant, as they have a chance to increase the odds of winning. And there is a mathematical explanation for why this is so. :D
     
  15. pianolady

    pianolady Monica Hart, Administrator Staff Member Piano Society Artist Trusted Member

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    I give up.
     
  16. Terez

    Terez New Member

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    Robert is close. He complicated it a bit too much, though. ;) And don't give up till you've had at least a couple of cups of coffee!
     
  17. Terez

    Terez New Member

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    Oh, and Chris...

    That, my dear, is exactly the point. ;)
     
  18. pianolady

    pianolady Monica Hart, Administrator Staff Member Piano Society Artist Trusted Member

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    Ok - I was going to give it one more shot, but knowing that Robert is right but too complicated makes it even more complicated. So this time, for sure I give up. Someone's got to do some work around here, anyway. :lol:
     
  19. Terez

    Terez New Member

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    Alright...since everyone apparently gave up, I'll go ahead and post the explanation. Here's a handy little chart, courtesy of Marilyn vos Savant, that assumes that you pick door #1:

    [​IMG]

    It should be pretty easy from this to see that, no matter which door you originally pick, once the host reveals a goat, then the remaining door (the one that neither you nor the host picked) gives you a 66.66...% chance of winning, while the door you picked in the first place remains at a 33.33...% chance.

    The top half of the chart and the bottom half are essentially the same thing, worded in two different ways, to further make clear the odds of winning if you stay, and the odds of winning if you switch.
     
  20. rsmullyan

    rsmullyan New Member Piano Society Artist

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    The explanation that Terez gave is due to Marylin Savant. I would explain it a different way: If you don't trade , then you will win just iin case you originally chose the corret door,right? But if you do trade, then you will win just in case you originally chose the wrong door( because then the host will open up the other wrong door, and you will trade to the right one). Now, isn't it twice as likely to originally choose one of the 2 wrong doors than the single right door? Thus if you do trade, you are twice as likely to wih than if you don't.

    I would like to know ( and I believe Terez also would) which of the two explanations you find more convincing-- Marylin's or mine? Please let us know!
     

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