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Posted by Plutonium on 20th July 2012 at 08:36 (4598 Views)
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JUL
20
2012
The Monty Hall problem
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While we're on the subject of probability, how about one of the most controversial problems in Mathematics, one that has caught out even professional mathematicians.

You are on a game show. The compere shows you three doors. Behind one door is a brand new car, behind two of the other doors are goats. You choose a door. The compere then opens one of the other two doors, to reveal a goat behind it. He then asks you if you want to stick with your original choice, or switch to the other remaining closed door.

Do you improve your chances (of getting the car) by switching to the other door?
Updated 20th July 2012 at 08:37 by Plutonium
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  1. Memnoch -
    JUL
    20
    2012
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    I see two valid solutions to this. It all depends on whether the compere knows what's behind the doors and whether he deliberately acts on that knowledge.

    If the host doesn't know, or doesn't make a deliberate informed choice, then your chances AFTER a goat is revealed are fifty fifty and will remain so if you swap. In this case, the host could have revealed the car, in which case you lose.

    Here are the possible outcomes in the "Compere doesn't know" scenario:

    1. You pick the car, compere reveals Goat 1, you stick = WIN
    2. You pick the car, compere reveals Goat 2, you stick = WIN
    3. You pick Goat 1, compere reveals Goat 2, you swap = WIN
    4. You pick Goat 2, compere reveals Goat 1, you swap = WIN
    5. You pick the car, compere reveals Goat 1, you swap = LOSE
    6. You pick the car, compere reveals Goat 2, you swap = LOSE
    7. You pick Goat 1, compere revelas Goat 2, you stick = LOSE
    8. You pick Goat 2, compere reveals Goat 1, you stick = LOSE
    9. You pick Goat 1, compere revelas the car = LOSE
    10. You pick Goat 2, compere reveals the car = LOSE

    So, there are four ways to win and six ways to lose. Two of those wins are sticks and two are swaps, so it makes no difference. If the car is not revealed, then you have four ways to win and four ways to lose and again stick or swap makes no difference. In this scenario, the odds change after the reveal because it might be the car, in which case you lose, and since sticking or swapping makes no difference in any case, your overall chances seem to be four out of ten.

    Except that they're not. Let's assume that because swapping makes no difference, you decide in advance not to swap no matter what. This means your winning or losing is determined by your original choice, which is a 1 in 3 chance. If on the other hand you decide in advance to swap if you get the opportunity to swap, then you can either:

    1. Pick the car = LOSE
    2. Pick Goat and reveal the car = LOSE
    3. Pick a goat and swap = WIN
    So it's still 1 in 3.

    Therefore, in the "Compere doesn't know" version of the game, your chances are 1 in 3 regardless of strategy.

    Turns out this agrees with your most likely intuition. You have no useful information, neither does anyone else and you have a simple choice with one winning chance and two losing chances.


    HOWEVER

    What if the compere DOES know, and decides in advance to reveal a goat no matter what you pick? This is much more interesting:

    Stick strategy:
    Pick the car = WIN
    Pick Goat 1 = LOSE
    Pick Goat 2 = LOSE
    And it's still 1 in 3. (Doesn't matter what the compere reveals because you have committed yourself to your original 1 in 3 choice)

    Swap strategy:
    1. Pick the car, reveal Goat 1 = LOSE
    2. Pick the car, reveal Goat 2 = LOSE
    3. Pick Goat 1, reveal Goat 2 = WIN
    4. Pick Goat 2, reveal Goat 1 = WIN
    Fifty-Fifty!


    So finally, we get to change the odds in our favour. If the compere deliberately reveals a goat, then your best strategy is to decide in advance to swap.

    Updated 20th July 2012 at 12:04 by Memnoch
  2. Memnoch -
    JUL
    20
    2012
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    The Monty Hall problem
    Plutonium
    one of the most controversial problems in Mathematics, one that has caught out even professional mathematicians.
    Not surprising since the problem isn't really mathematical. By that I mean there aren't any obscure techniques or complex relationships. It's not really about number at all except at a very basic level, it's mainly a matter of paying attention and considering every possibility. If people don't find the solution, it's only because they didn't consider all the possible outcomes. In this regard it's similar to the three card problem you posed earlier.
  3. Plutonium -
    JUL
    20
    2012
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    Memnoch

    Surely it is axiomatic that the compere knows the location of the car; otherwise he lacks control over the show. So the problem assumes that the compere has this knowledge. Otherwise, when he opens one of the doors, he could inadvertently reveal the car. Show's over.

    I suggest that your analysis, assuming the compere knows, is missing something important. A door has already been opened to reveal a goat. It matters not whether that is 'Goat1' or 'Goat2'. It's not like the three-card trick, where it does matter which of the red sides you've got, because that determines if the other side is blue. All you know is that behind one of the two remaining closed doors is a goat, and behind the other one is a car.

    So do you stick or swap?
  4. Plutonium -
    JUL
    20
    2012
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    BTW, Maths doesn't necessarily involve 'obscure techniques or complex relationships'. Anything connected with the significance of or manipulation of numbers is mathematical, from the solution to 1 + 1, up to the most complex calculus.

    Sudoku is often described as a mathematical puzzle, but I suggest that it isn't, because the solution does not depend on the mathematical relationship between the figures. You could equally well play Sudoku with a grid of letters, or meaningless symbols, the rules being the same: no symbol can be repeated in any row, column or square.

    OTOH, Kakuro, is mathematical, because it depends on the relation between the numbers. The rows and columns have to add up to the number shown.

    Probability is mathematical because it is based on theories of probability as expressed as a numerical value between 0 and 1.
  5. Memnoch -
    JUL
    21
    2012
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    Plutonium
    Memnoch

    Surely it is axiomatic that the compere knows the location of the car; otherwise he lacks control over the show. So the problem assumes that the compere has this knowledge. Otherwise, when he opens one of the doors, he could inadvertently reveal the car. Show's over.
    Hang on there. You're the one who presented it as a mathematics problem. Do I have to explain to you what an axiom is, or do you in fact already know that the compere's knowledge is not REMOTELY "axiomatic"? My analysis was based on the information given, which was incomplete, hence I analysed both the "compere knows" and the "compere doesn't know" versions. This is not mere pedantry: If you are presented with an ambiguous problem on your finals, you will have just cause to be miffed

    [Apologies to Memnoch - I tried to reply to your comment but ended up inadvertently editing it - this was not deliberate - I'm not used to the blog format - Plutonium]

    Updated 21st July 2012 at 09:20 by Plutonium
  6. Memnoch -
    JUL
    21
    2012
    Memnoch's Avatar
    Plutonium
    BTW, Maths doesn't necessarily involve 'obscure techniques or complex relationships'. Anything connected with the significance of or manipulation of numbers is mathematical, from the solution to 1 + 1, up to the most complex calculus.
    Depends on your level. To a layperson, arithmetic is mathematics. To a mathematician, mathematics is everything except arithmetic. As you are presenting from the perspective of a higher student, I presumed you would be in the latter category. However, yes, strictly speaking it is mathematical, but it's not degree level mathematics, even if some professors have got it wrong. It's actually laughably simple if you stick to the rule I posited and analyse everything because the analyses themselves are not, numerically speaking difficult. They are in fact much simpler than calculating the odds of winning at the blackjack table.

    Sudoku is often described as a mathematical puzzle, but I suggest that it isn't, because the solution does not depend on the mathematical relationship between the figures. You could equally well play Sudoku with a grid of letters, or meaningless symbols, the rules being the same: no symbol can be repeated in any row, column or square.
    I agree completely and you remind me of a recent "discussion" with a colleague who told me she can't do Sudoku puzzles because she's rubbish at anything mathematical.

    OTOH, Kakuro, is mathematical, because it depends on the relation between the numbers. The rows and columns have to add up to the number shown.
    True, although I can't find an algorithm so in practice the solution is not purely mathematical: I usually solve those, at least partially, by trial and error.
    Probability is mathematical
    I never said it wasn't. I said - or at least tried to imply - that the style of logic required to solve this type of problem doesn't involve any mathematics beyond the trite strategy of listing all possible outcomes and noticing whether the wins outnumber the losses. Again yes, it's mathematics, but not a high level - it's simply counting.

    Updated 21st July 2012 at 01:12 by Memnoch
  7. Evil -
    JUL
    21
    2012
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    I used to like maths and game shows. Now I hate both.
  8. Plutonium -
    JUL
    21
    2012
    Plutonium's Avatar
    Memnoch
    Hang on there. You're the one who presented it as a mathematics problem. Do I have to explain to you what an axiom is, or do you in fact already know that the compere's knowledge is not REMOTELY "axiomatic"? My analysis was based on the information given, which was incomplete, hence I analysed both the "compere knows" and the "compere doesn't know" versions. This is not mere pedantry: If you are presented with an ambiguous problem on your finals, you will have just cause to be miffed
    Well, Memnoch, I will take that on board, should I refer to the Monty Hall problem again, somewhere else. I will have to state clearly that the compere knows where the car is.

    I would have personally thought that it is self-evident that the compere knows where the car is, for the Monty Hall test to work (ie asking the player to stick or swap).

    Think about it. If he opens a door and the car is behind it, then he cannot ask the player if they want to stick or swap.

    But even if the Monty Hall test (stick or swap) is not used, the compere still needs to know where the car is.

    Let's say the player opens a door and there is a goat behind it. The compere says, "too bad, you should have opened this door!" and opens a door with another goat behind it. That just makes him look stupid, doesn't it?

    Beside which, is the compere's knowedge really relevant?

    If the compere knows, then we can be sure that he will open a door with a goat behind it. You then have two doors left, one with a car behind it, one with a goat behind it. You just don't know which is which. Let us call this position X.

    If the compere doesn't know, then there is a chance that he will open the door with the car behind it, in which case there is no point in continuing the game - the player has lost.

    However, if he opens a door and there is a goat behind it, then we are back at position X. Which is the original position in the Monty Hall problem. The compere's knowledge, or ignorance, of which door the car is behind, is irrelevant to whatever the probability is.
  9. Plutonium -
    JUL
    21
    2012
    Plutonium's Avatar
    Anyway, Memnoch added some other comments, which I managed to lose because I thought I was replying to his reply but was actually editing it. Apologies again, hopefully I'll get the hang of this.

    Basically, Memnoch, you were arguing that it is important which goat it is. Please re-post your arguments if you have time.

    My view is this. At position X, all we know is that there are two closed doors left - one with a goat behind it and one with a car behind it. I can't see that it matters whether the goat is Goat1 or Goat2, any more than it matters whether the car is a Rolls Royce or a Mercedes.

    In the case of the three-card trick, it does matter whether you have side R1, R2 or R3. Because only R3 is blue on the other side.

    In Monty Hall, whether you've got Goat1 or Goat2, it doesn't matter - you've lost.

    Or look at it another way. In the three-card trick, supposing the cards were marked. If you saw that the card was marked R1 or R2, then you would know the other side was red. If you saw that the card was marked R3, you would know that the other side was blue.

    In the Monty Hall problem, suppose the goats were also marked. When the compere opens a door to reveal Goat2 behind it, how does that changes the chances of winning the car in any way, from not knowing if it's Goat1 or Goat2?

    Stated differently.

    If you don't know which goat is behind the open door, then you have this:

    One closed door has a car behind it.
    The other closed door has either Goat1 or Goat2 behind it.
    The open door has either Goat1 or Goat2 behind it.

    If the goat is marked Goat2, then you have this:

    One closed door has a car behind it.
    The other closed door has Goat1 behind it.
    The open door has Goat2 behind it.

    I can't see how knowing which goat is behind the open door, changes the probabilities of getting the door with the car behind it.

    Updated 21st July 2012 at 09:39 by Plutonium
  10. Plutonium -
    JUL
    21
    2012
    Plutonium's Avatar
    Memnoch
    Depends on your level. To a layperson, arithmetic is mathematics. To a mathematician, mathematics is everything except arithmetic. As you are presenting from the perspective of a higher student, I presumed you would be in the latter category. However, yes, strictly speaking it is mathematical, but it's not degree level mathematics, even if some professors have got it wrong.
    I'm a 'higher student', yes, but I also have to pitch the blog to a level that will appeal to a wider audience. Stephen Hawking was advised that every formula he included in his book "A brief history of Time" would reduce his sales by half. If I started discussing the derivative of f(x) = 3e^x/6, I doubt if anyone would bother reading it.

    Incidentally, to a mathematician, arithmetic is mathematics, albeit at its most simple level. Although Arithmetic may seem simple, until you try doing it in base 2, base 8, base 16 or another base - there's nothing particularly special about base 10.

    It's actually laughably simple if you stick to the rule I posited and analyse everything because the analyses themselves are not, numerically speaking difficult. They are in fact much simpler than calculating the odds of winning at the blackjack table.
    Hmm ... it's not quite as simple as it looks, I would suggest. The trick with probability is that what may seem obvious is not actually correct.

    I agree completely and you remind me of a recent "discussion" with a colleague who told me she can't do Sudoku puzzles because she's rubbish at anything mathematical.

    True, although I can't find an algorithm so in practice the solution is not purely mathematical: I usually solve those, at least partially, by trial and error.I never said it wasn't. I said - or at least tried to imply - that the style of logic required to solve this type of problem doesn't involve any mathematics beyond the trite strategy of listing all possible outcomes and noticing whether the wins outnumber the losses. Again yes, it's mathematics, but not a high level - it's simply counting.
    I would love to know if there is an algorithm for Kakuro. I use trial and error myself.

    Updated 21st July 2012 at 09:50 by Plutonium
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