Probability 101: Monty Hall - Page 4

Probability 101: Monty Hall

This is a discussion on Probability 101: Monty Hall within the Off Topic & Humor Discussion forums, part of the The Back Porch category; Originally Posted by Hopyard Yes, but IF you don't know that you picked the quarter and choose "don't switch" (an option not given in your ...

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Thread: Probability 101: Monty Hall

  1. #46
    VIP Member Array JonInNY's Avatar
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    Quote Originally Posted by Hopyard View Post
    Yes, but IF you don't know that you picked the quarter and choose "don't switch" (an option not given in your example) you win.
    Yes, Hopyard, you are correct there. Obviously you do not know what you picked... I was just using the coins as an example.

    The original question asked, "given the opportunity", do you switch? The answer needs to be yes, because the original odds do not change... it's just that more information is provided.

    I know it does not make sense on the surface, and I didn't get it either, and actually used the coins as a metaphor so I could physically see it. But once I worked it out, I saw the odds were in my favor if I switched!

    Jon
    "Democracy is two wolves and a lamb voting on what to have for lunch; Liberty is a well-armed lamb contesting the vote."
    -- Benjamin Franklin


  2. #47
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    re: kazaerexys

    Quote Originally Posted by kazzaerexys View Post
    Hopyard, I want to try to answer this, and please understand that I am in no way, shape, or form, trying to pick on you.
    Not to worry. I don't think you are picking on me.

    The fact of the matter is, most people simply don't have a gut feeling for probabilities. Probabilities are useful as expressions of long-term trends in outcomes. If you roll a 6-sided die a couple hundred times, about one-sixth of your rolls will be 1, about one-sixth will be 2, and so on.
    Of course. The question in my mind is whether or not this is the analogy for the goat -BP game.

    Mathematically, we say that on your very next roll of that die, you have a 1/6th chance (or probability) of getting, say, a 6. That's the definition of probability.

    Yes, of course.

    You, on the other hand, like a lot of people, look at it from the very direct point of view that either you get a 6 or you don't.
    Don't think so. I wouldn't do that with the 6 sided die. I may or may not be making that error in my analysis of this problem, but I don't see the comparison with the die; I see the correct analog as the coin toss. I think I know what you are driving at though.

    Two outcomes. And because most people really don't get probabilities at a gut level, everything tends to look to them like "certainly true", "certainly false", or "50/50"---that is, probability 100%, 0%, or 50%---even though that's not what the math says about the situation.
    I'll admit to the possibility that I am viewing this puzzle falsely as one involving dichotomous choices, and I'll admit that I might not be sharp enough to see that it isn't about independent dichotomous events at all, even after you guys have pointed that issue out. I'm just not quite convinced by the example given by JoininNY because an option seems to me is being omitted, there are 4 options not three as enumerated (I think).

    Let me give another example---you're playing poker and you have four of a kind. You are about to rake in a huge pot against one guy still betting into you. Maybe he has a royal flush and maybe he doesn't. Those are the possibilities, but it is certainly not a 50% chance that he has a royal flush.
    AGREE

    In fact, I think the odds of a royal flush are about 650,000:1 against, but if he does pull that royal flush, those low odds aren't going to be much consolation to you...
    Agree again.

    By the way (and this may be a rather tangential aside), these are not just simple little game-playing consequences.
    Couldn't agree more.

    The fact that people in positions of power and authority don't understand odds bites folks in the butt all the time. What were the odds of a plane being downed by the volcanic ash over Europe this past week? Pretty slim; maybe even infinitesimal. But because of the fact that EU functionaries can't tell the difference between "some small risk of disaster" and "near certainty of disaster," the EU bureaucracy made a decision that cost the airline industry a couple billion dollars.

    </math_rant>
    Yes, but I think the policy issue really wasn't one of odds. Again, tangentially, you wouldn't want to be either the passenger or the politician involved in the decision if one went down. I think your point is well intended and well made however. These are indeed not trivial problems for fun and games and are best left to the mathematical statisticians when money and lives are at stake.

    Now back to the puzzle-- is there not a choice being left out in Joinin's analysis? Or is what you guys are driving at the business end of "degrees of freedom, n-1"???

    I know I'm over my head when I play with mathematicians and statisticians, but I'm not finding the plain English explanations convincing---though if you guys are in fact real world working statisticians I will have no choice but to bow to your professional opinion. I guess you have no obligation to convince a "dunce." Just as Einstein had no obligation to convince anyone outside the world of physics--only the guy calculating yield-- that his unintuitive conclusions were correct.

  3. #48
    VIP Member Array JonInNY's Avatar
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    Quote Originally Posted by Hopyard View Post
    Now back to the puzzle-- is there not a choice being left out in JoinNY's analysis?
    The "4th choice" as you put it (and as I think I understand it), is to do nothing and keep the door you picked originally. That's an acceptable choice, of course, but the odds are still better if you switch.

    Try and play the coin game I mentioned from every possibility, and write down the choices and odds. And let's not forget the original question: "Are your odds better if you switch?"

    It not obvious and it took me a while to figure out why.
    "Democracy is two wolves and a lamb voting on what to have for lunch; Liberty is a well-armed lamb contesting the vote."
    -- Benjamin Franklin

  4. #49
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    You all are most certainly entitled to be in the 50/50 "camp," but, to be blunt, it's simply wrong. This is math, not opinion - you could be in the 2+2=5 "camp," but that doesn't mean that "camp" has any sort of validity. Do the math. Do the coin example. It is a mathematical CERTAINTY that you have a 1/3 chance of winning if you stay, and a 2/3 chance of winning if you switch. There can be no differences of opinion (well, there can be, but in math there actually IS one, and in this case ONLY one, correct answer).
    A man fires a rifle for many years, and he goes to war. And afterward he turns the rifle in at the armory, and he believes he's finished with the rifle. But no matter what else he might do with his hands - love a woman, build a house, change his son's diaper - his hands remember the rifle.

  5. #50
    VIP Member Array JonInNY's Avatar
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    Quote Originally Posted by OPFOR View Post
    well, there can be, but in math there actually IS one, and in this case ONLY one, correct answer).
    There is no correct answer to "what is the last digit of pi?" Actually, no answer at all. But I digress...

    Thanks again OPFOR for your explanation. I fully understand now, especially with the help of my coin metaphor. I found it much easier to work with physical items!
    "Democracy is two wolves and a lamb voting on what to have for lunch; Liberty is a well-armed lamb contesting the vote."
    -- Benjamin Franklin

  6. #51
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    Quote Originally Posted by JonInNY View Post
    There is no correct answer to "what is the last digit of pi?" Actually, no answer at all. But I digress...
    Ah, but there IS the mathematical concept of infinity... You can't simply be in the "there is no Pi" camp and be mathematically correct.

    Oh, and I'm not a mathematician or a statistician - I'm a knuckle dragger who happens to play poker and who also taught himself to count cards in Blackjack (for fun only - as a lone player without a massive bankroll and with all the "counter-counting" practices in place these days it's basically impossible to gain any advantage by counting). The concepts of conditional probability (as kazz mentioned) are the foundations of these games (and, of course, of a lot of other things...)
    A man fires a rifle for many years, and he goes to war. And afterward he turns the rifle in at the armory, and he believes he's finished with the rifle. But no matter what else he might do with his hands - love a woman, build a house, change his son's diaper - his hands remember the rifle.

  7. #52
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    re: OPFOR never mind

    Quote Originally Posted by OPFOR View Post
    You all are most certainly entitled to be in the 50/50 "camp," but, to be blunt, it's simply wrong. This is math, not opinion - you could be in the 2+2=5 "camp," but that doesn't mean that "camp" has any sort of validity. Do the math. Do the coin example. It is a mathematical CERTAINTY that you have a 1/3 chance of winning if you stay, and a 2/3 chance of winning if you switch. There can be no differences of opinion (well, there can be, but in math there actually IS one, and in this case ONLY one, correct answer).
    You are right there is no room for a difference of opinion. It is math. And it has to make sense, which it may well be doing even though my poor old mind isn't grasping what you guys are trying to convey about how you reached your conclusion.

    That's why I gave the example from physics of the somewhat counterintuitive energy mass relationship. I don't need to understand it to know it is correct. Same here. I'm going to have to take your and Joinin's word that your explanations and examples are correct, because I do not understand it.

  8. #53
    Senior Member Array Lewis128's Avatar
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    I'm reposting this link because there was a typo previously.
    If you watch the whole thing and still insist the odds are 50:50, I give up.

    YouTube - The Monty Hall Problem Explained

  9. #54
    VIP Member Array JonInNY's Avatar
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    Quote Originally Posted by Hopyard View Post
    I'm going to have to take your and Joinin's word that your explanations and examples are correct, because I do not understand it.
    Hang in there Hopyard. I couldn't get it either. Took me a day and many pushing around of coins. You may have a "Eureka" moment in the middle of the night!
    "Democracy is two wolves and a lamb voting on what to have for lunch; Liberty is a well-armed lamb contesting the vote."
    -- Benjamin Franklin

  10. #55
    VIP Member Array JonInNY's Avatar
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    Quote Originally Posted by Lewis128 View Post
    I'm reposting this link because there was a typo previously. If you watch the whole thing and still insist the odds are 50:50, I give up.

    YouTube - The Monty Hall Problem Explained

    That's a great explanation, and hopefully will convert the 50/50'ers. I like the one lump or two analogy!
    "Democracy is two wolves and a lamb voting on what to have for lunch; Liberty is a well-armed lamb contesting the vote."
    -- Benjamin Franklin

  11. #56
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    Whew! It's time for another
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  12. #57
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    Good link. Is there anyone still not getting it? 'Cuz I got another one...
    A man fires a rifle for many years, and he goes to war. And afterward he turns the rifle in at the armory, and he believes he's finished with the rifle. But no matter what else he might do with his hands - love a woman, build a house, change his son's diaper - his hands remember the rifle.

  13. #58
    VIP Member Array JonInNY's Avatar
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    Quote Originally Posted by OPFOR View Post
    Cuz I got another one...
    Bring it on!
    "Democracy is two wolves and a lamb voting on what to have for lunch; Liberty is a well-armed lamb contesting the vote."
    -- Benjamin Franklin

  14. #59
    Distinguished Member Array Paymeister's Avatar
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    10 years in college, a Master's in Public Health Epidemiology, and a Certified Payroll Professional.

    You were right, I was wrong. I'm sorry. Seriously - apologies for pushing the wrong way. Sigh.

    The key here is that it is NOT a second problem: the car stayed put. Good YouTube explanation - thanks for the link.
    Recently updated website: http://www.damagedphotorepair.com

  15. #60
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    No worries, Paymeister, it is counter-intuitive, and it seems obvious at first glance. At least you've figured out that "dice have no memory," although it may have cost you some cash to learn that one!

    I'll post up a new one tomorrow.
    A man fires a rifle for many years, and he goes to war. And afterward he turns the rifle in at the armory, and he believes he's finished with the rifle. But no matter what else he might do with his hands - love a woman, build a house, change his son's diaper - his hands remember the rifle.

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