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  1. #1
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    the price is right

    I remember an assertion, can't find it right now, but I was reminded of it while skimming the Absence of evidence thread , it went something like:

    You are playing the price is right. You have three curtains to choose from, A, B, and C and one has something you want, the other two have zonkers.

    you choose curtain A.

    Bob (or drew) opens up curtain B, and shows you a zonker.

    Bob then says, do you want to stick with curtain A or choose curtain C?

    Now according to what I was reading, you should automatically switch to curtain C.

    Does that make logical sense to anyone? It doesn't make sense to me, currently, though I don't see how it would hurt to switch. The discussion seemed to imply that your odds were better if you switched.

    ---------- Post added at 10:11 AM ---------- Previous post was at 09:00 AM ----------

    Ah, found a link
    http://www.theproblemsite.com/treasu.../door_hint.asp, apparently it is let's make a deal, not price is right. Pick me Monty!!
    Last edited by daveId; December 22nd, 2010 at 07:09 AM.

  2. #2
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    Re: the price is right

    Sounds like an absurd notion to me.

    Your odds of winning changed the moment they revealed curtain B to 50% from 33.3%. Changing curtains does nothing to change that one way or another.
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  3. #3
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    Re: the price is right

    You DO want to change curtains. Doing so makes the odds 66% that you picked the wrong one to 66% that you picked the right one.

    If you originally picked the wrong curtain, switching the curtain will make you a winner.

    If you originally picked the right curtain, switching the curtain will make you a loser.

    So what are the odds that you originally picked the wrong curtain? 66%

    So what are the odds you will win if you switch curtains? Again, 66%.

    So you want to switch curtains when offered.
    Last edited by mican333; December 22nd, 2010 at 07:54 AM.

  4. #4
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    Re: the price is right

    Quote Originally Posted by mican333 View Post
    So what are the odds that you originally picked the wrong curtain? 66%

    So what are the odds you will win if you switch curtains? Again, 66%.

    So you want to switch curtains when offered.
    Mican, I can't say I follow the logic. If it's a 66% probability that I picked the wrong curtain, how is it not true that switching curtains will give me a 66% chance of also picking the wrong curtain?

    BTW, I hated probability. If you can explain this to me in a way that makes sense, I'll rep you. Consider that a challenge.
    "And that, my lord, is how we know the Earth to be banana-shaped." ~ Monty Python


  5. #5
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    Re: the price is right

    This is the "Monty Hall" Problem [1]. It's fairly well known exactly because the right answer is counter-intuitive.

    Monty Hall was the host of "Let's Make a Deal", not "The Price is Right".

    A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three. The probability of winning by staying with the initial choice is therefore 1/3, while the probability of winning by switching is 2/3.

    [1] http://en.wikipedia.org/wiki/Monty_Hall_problem
    "... freedom is not, as we are told, a liberty for every man to do what he lists but a liberty to dispose, and order as he lists, his person, actions, possessions, and his whole property, within the allowance of those laws under which he is, and therein not to be subject to the arbitrary will of another, but freely follow his own." -- John Locke, Second Treatise on Government

  6. #6
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    Re: the price is right

    Quote Originally Posted by ladyphoenix View Post
    Mican, I can't say I follow the logic. If it's a 66% probability that I picked the wrong curtain, how is it not true that switching curtains will give me a 66% chance of also picking the wrong curtain?

    BTW, I hated probability. If you can explain this to me in a way that makes sense, I'll rep you. Consider that a challenge.
    I realized I wrote something incorrectly in my post which might be adding to your confusion.

    The simplest way to demonstrate this is to go through all of the possibilities.

    So let's say that Curtain A has the car and Curtain B & C has the Zonker.

    If you don't switch, the odds are 1 out of 3 that you will pick A and win.

    But if you do switch:

    If you originally pick Curtain A, host reveals B or C (either will do since they both have Zonkers), and you switch to the other curtain that has a Zonker. LOSE

    If you pick Curtain B, host reveals C (the only zonker curtain remaining) and you switch to A. WIN

    If you pick Curtain C, host reveals curtain B (the only zonker curtain remaining) and you switch to A. WIN.

    So as you can see, by switching curtains, the odds are two out of three that you will win.

    ---------------------------------------

    The part of the equation that people tend to not factor in is that the curtain the host reveals is not decided prior to the game but is decided after the first pick and he will reveal a zonker, thus eliminating the possibility of picking that particular zonker if you switch. So the only possibility if you switch is that if you originally picked the car, you will switch to a zonker and if you originally picked a zonker you will switch to a car. And since the odds are 66% that originally picked the zonker, switching makes it now 66% that you win the car.

  7. #7
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    Re: the price is right

    When you make your original choice there is one chance in three that you chose the right one. When one of them is eliminated each of the remaining two curtains has an equal chance of being the right one. This is true regardless of which one you have chosen. So switching to the other one doesn't affect your chances of winning.

  8. #8
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    Re: the price is right

    Quote Originally Posted by theophilus View Post
    When you make your original choice there is one chance in three that you chose the right one. When one of them is eliminated each of the remaining two curtains has an equal chance of being the right one.
    But which of the three curtains he eliminates may be influenced by your original pick so it's a different situation and if he eliminated a curtain prior to your original pick (which would be a 50-50 chance). Him eliminating a curtain after your pick tells you something about your pick and therefore does not make it a straight 50-50 in whether you should switch or not.

    Switching curtains means that if you picked the winning curtain, you will switch to the remaining losing curtain. And if you picked a losing curtain, you will switch to the winning curtain.

    So if you had a one in three chance of winning with your original pick, switching will mean you now have a one in three chance of losing and therefore a two in three chance of winning.

  9. #9
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    Re: the price is right

    wow, looks like this was discussed here 5 years ago
    http://www.onlinedebate.net/forums/s...y-Hall-Problem.

    mican, I like your explanation in the second part of post 6, still thinking about that.

  10. #10
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    Re: the price is right

    Yep, the wiki article enlightened me.

    The trick of it is the behavior of the show host.
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  11. #11
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    Re: the price is right

    Quote Originally Posted by mican333 View Post
    The part of the equation that people tend to not factor in is that the curtain the host reveals is not decided prior to the game but is decided after the first pick and he will reveal a zonker, thus eliminating the possibility of picking that particular zonker if you switch. So the only possibility if you switch is that if you originally picked the car, you will switch to a zonker and if you originally picked a zonker you will switch to a car. And since the odds are 66% that originally picked the zonker, switching makes it now 66% that you win the car.
    I get it right up to the bolded. How does it follow that it must be 66% that you win the car?
    "And that, my lord, is how we know the Earth to be banana-shaped." ~ Monty Python


  12. #12
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    Re: the price is right

    Quote Originally Posted by ladyphoenix View Post
    I get it right up to the bolded. How does it follow that it must be 66% that you win the car?
    Because it was originally 66% that you didn't pick the car. Since switching means you will pick the opposite of your original pick, it will be 66% that you will switch to the car.

  13. #13
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    Re: the price is right

    Quote Originally Posted by mican333 View Post
    Because it was originally 66% that you didn't pick the car. Since switching means you will pick the opposite of your original pick, it will be 66% that you will switch to the car.
    How can you not say it is 66% if I stick with my original pick and 66% if I don't? Why does switching have anything to do with it? (Maybe I should take the time to read the wiki article.)
    "And that, my lord, is how we know the Earth to be banana-shaped." ~ Monty Python


  14. #14
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    Re: the price is right

    Quote Originally Posted by Sigfried View Post
    The trick of it is the behavior of the show host.
    Exactly. And what door he reveals is determined by your original pick and likewise gives you information about whether you should switch. If the door he opened was decided prior to your original pick, then him opening that door wouldn't give any additional information.

    ---------- Post added at 02:15 PM ---------- Previous post was at 01:57 PM ----------

    Quote Originally Posted by ladyphoenix View Post
    How can you not say it is 66% if I stick with my original pick and 66% if I don't? Why does switching have anything to do with it? (Maybe I should take the time to read the wiki article.)
    Like I said, I mistyped earlier. It's 33% if you stick with your pick and 66% if you switch.

  15. #15
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    Re: the price is right

    Quote Originally Posted by mican333 View Post
    Because it was originally 66% that you didn't pick the car. Since switching means you will pick the opposite of your original pick, it will be 66% that you will switch to the car.
    But if you had chosen the other remaining door wouldn't the odds be exactly the same? When you made the first choice every door had a 66% chance of being the wrong one. Each of the two remaining doors has a 50% chance of being the right one regardless of which one you have selected.

  16. #16
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    Re: the price is right

    Quote Originally Posted by theophilus View Post
    But if you had chosen the other remaining door wouldn't the odds be exactly the same?
    There is no other remaining door. The host opened it. The only possible outcome is that you will switch from a winner to a loser or switch from a loser to a winner.

    And since odds are 66% that you chose loser with your original pick, if you switch odds are 66% that you switched to a winner.

    Quote Originally Posted by theophilus View Post
    When you made the first choice every door had a 66% chance of being the wrong one. Each of the two remaining doors has a 50% chance of being the right one regardless of which one you have selected.
    But the host opened a losing door before offering you the option of switching.

    The thing to keep in mind is that the host did not open a door at random - he opened a losing door. So him opening a losing door tells you something.

  17. #17
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    Re: the price is right

    For anyone still struggling I will also try to explain.

    So let's say you picked door one, and the host opened door 3.

    What happens most of the time:

    Your original pick of door one was the wrong choice- 66% probability. (33% probability you were correct.) A common mistake made at this point is that the host revealing the other door increases your probability to 50%. (Of being correct) But, how does him showing a door increase your original probability. If I closed my eyes when he revealed the door the probability would remain at 33%. So how does me knowing increase my chances? It doesn't.
    Because there is a 66% probability that I was wrong, it is the safer bet to switch. Because I already know what door one of the wrong ones is at, that eliminates one. And the one I originally picked is also eliminated because there is a 66% chance that it was wrong. This gives switching a 66% probability of being correct.
    Udabindu yathāpi pokkhare
    Padume vāri yathā na lippati,
    Evaṃ muni no palippati
    Yadidaṃ diṭṭhasutaṃ mutesu vā.

  18. #18
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    Re: the price is right

    Quote Originally Posted by Soren View Post
    A common mistake made at this point is that the host revealing the other door increases your probability to 50%.
    This really is the big hinging point. Probability is calculated based on information known at the time the calculation is made.

    New information that comes to light after a calculation is made changes probability results for new calculations, but does not change the probabilities of the original calculations.

    At no point is the equation ever able to equal 50% because at no time are you ever considering just 2 doors. You're initially considering 3 closed doors, and then you're considering 2 closed doors and 1 open door, which is still 3 doors. So, all probabilities are always going to be evenly divisible by 3.

    Quote Originally Posted by Monty Python and the Holy Grail
    Then shalt thou count to three, no more, no less. Three shall be the number thou shalt count, and the number of the counting shall be three. Four shalt thou not count, neither count thou two, excepting that thou then proceed to three. Five is right out.
    The only way you can get to 50% probability is if you could go back in time to before making your initial selection, taking the knowledge of which door is opened to reveal the goat with you. Unfortunately, time travel in the reverse direction isn't currently possible outside of Hollywood.
    "... freedom is not, as we are told, a liberty for every man to do what he lists but a liberty to dispose, and order as he lists, his person, actions, possessions, and his whole property, within the allowance of those laws under which he is, and therein not to be subject to the arbitrary will of another, but freely follow his own." -- John Locke, Second Treatise on Government

  19. #19
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    Re: the price is right

    I've read this problem many times. What I don't get is why the odds are 66% either way. In actuality you are only choosing between two doors since you will always be shown a zonker your only choices are ever your door and the remaining closed door. The third door is irrelevant in the problem except as a distraction.

  20. #20
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    Re: the price is right

    your initial guess had a 2/3 chance of being wrong.

    if you picked zonk #1 originally, and then switch, you will win (monty just showed you zonk #2)

    if you picked zonk #2 originally, and then switch, you will win (monty just showed you zonk #1)

    if you picked the prize originally, and switch, you lose (monty just showed you a random zonk).

    2 win scenarios vs 1 lose IF you switch =%66

 

 
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