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Thread: Do you know the correct answer?

  1. #771
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    Default Re: Do you know the correct answer?

    Quote Originally Posted by coelcanth View Post
    no, you and Nick are both stuck on the same point..

    revealing a goat in the second round does nothing to affect the probability of a pick that was made in the past

    thinking about the reveal as 'removing' that goat from play is probably the wrong way to go about it and that's what's confusing you.
    revealing the specific location of the goat really makes no difference (we already knew Monty had to have AT LEAST ONE goat, right?)
    the particular door that it is in has no effect on whether Monty has the prize or not

    if we think of Monty's two doors as his entire hand we can say that his entire hand has a 2 in 3 chance of containing the prize
    opening the one door means that entire 2 in 3 chance now rests on the only unopened door in Monty's hand

    that is why his last door has a 2/3 chance and your one earlier pick has only a 1/3 chance of winning
    Go back and read my post. Our last sentences are nearly word-for-word identical.

    I am in not in any agreement with Nick in this.

    So it’s either my piss-poor writing skills or your reading comprehension.

    I’ll tell you what, how ‘bout we split that 50/50?

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    Default Re: Do you know the correct answer?

    33%, best offer!
    There's a lot of things they didn't tell me when I signed on with this outfit....

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by George Jung View Post
    33%, best offer!
    Oh no…. I’m switching to the other door!

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    Quote Originally Posted by ShorelineJohn View Post
    The show has known about this for most of the show, I watched a show years ago where Monty Hall himself talked about it. And the internet was decades away! The idea the show has a problem with this is silly, they have loved it. The show is for advertisers to highlight their products while the audience is watching a show. This is the stuff dreams are made of for those guys!
    I do think the equation is to subtract the highest priced prize that is less than the grand prize from the two remaining choices.

    Also bear in mind that in the real Let's Make a Deal!, there's a whole lot more going on.

    Monty sometimes offers cash up front to entice the contestant to switch (or not switch).

    The lovely Carol Merrill pops up out of a trap at random with a cart carrying a box that might or might not contain something of value, and that gets figured into the equation, too: "Stick with the door you selected? Or switch to the box that the lovely Carol Merrill has for you?"

    Sometimes, it's even, "You switch doors? Great! I've got another deal for ya -- the lovely Carol Merrill has just brought this box on stage. Would you be interested in switching to that?"

    The "Monty Hall Problem" is the stripped-down minimal example for analysis.
    You would not enjoy Nietzsche, sir. He is fundamentally unsound. — P.G. Wodehouse (Carry On, Jeeves)

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    Default Re: Do you know the correct answer?

    How can the probabilities of choosing between 2 different doors be 30% and 50%?

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    Default Re: Do you know the correct answer?

    I apologise for not having read everything since I originally posted, but am somewhat short of time.

    However, respondents seem to have missed my point. Perhaps I was not clear enough.

    Contestant chooses a door. Monty opens one of the two remaining doors to reveal a goat.

    Statistics say that Contestant can double his chances of winning the car by switching.

    But it doesn't seem to matter which door the contestant originally chose.

    He chose door 1; his chances double by switching to door 2.

    He chose door 2; his chances double by switching to door 1.

    This is the peculiarity i would like to see explained.

  7. #777
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    Default Re: Do you know the correct answer?

    Quote Originally Posted by miketaylor View Post
    I apologise for not having read everything since I originally posted, but am somewhat short of time.

    However, respondents seem to have missed my point. Perhaps I was not clear enough.

    Contestant chooses a door. Monty opens one of the two remaining doors to reveal a goat.

    Statistics say that Contestant can double his chances of winning the car by switching.

    But it doesn't seem to matter which door the contestant originally chose.

    He chose door 1; his chances double by switching to door 2.

    He chose door 2; his chances double by switching to door 1.

    This is the peculiarity i would like to see explained.
    Because all doors are equal. They all have an equal chance to win.

    Once you have chosen a door, you have a 1 in three chance of winning. Monty has a 2 in 3 chance. You know one of his is wrong from the start. Your choice of door doesn’t affect the odds. And he can always show you one door with a goat because he HAS to have one, and he knows which it is.
    Tom

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    Default Re: Do you know the correct answer?

    Simple explanation

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    Default Re: Do you know the correct answer?

    Comprehensive lecture by a PhD at Harvard, first 25 minutes of his talk.



    Edited to add, the lecturer is Joe Blitzstein, Professor of the Practice in Statistics, Harvard University, Department of Statistics
    Last edited by oldcodger; 06-16-2022 at 04:42 AM.

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    Default Re: Do you know the correct answer?

    What seems to be the consensus amongst some here is that the set of probabilities of first choice (1/3) is carried over somehow to the second set of probabilities in the second choice (1/2).

    Nick (and I) can not see that they are connected in anyway. How are the 2 sets of probabilities connected?

    To my mind it doesn't matter a damn which door I chose on first choice, or which door is the goat reveal, the second choice of 2 doors is completely unrelated to the first choice of 3 doors, yet some here say they are entangled some how?

    So what's the mechanism?

  11. #781
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    Default Re: Do you know the correct answer?

    Quote Originally Posted by lupussonic View Post
    What seems to be the consensus amongst some here is that the set of probabilities of first choice (1/3) is carried over somehow to the second set of probabilities in the second choice (1/2).

    Nick (and I) can not see that they are connected in anyway. How are the 2 sets of probabilities connected?

    To my mind it doesn't matter a damn which door I chose on first choice, or which door is the goat reveal, the second choice of 2 doors is completely unrelated to the first choice of 3 doors, yet some here say they are entangled some how?

    So what's the mechanism?
    The are connected because the “game” was never reset. Monty didn’t open a door randomly leaving you with new choices. You new choice is really do I stick with my one door, or do I choose both doors that Monty had. 796 is a good explanation.

    What information did you really gain by him showing you what was behind one of his doors? If, before opening a door to reveal a goat Monty asked if you wanted his two doors or stick with your 1 door would you stay or switch? You would switch, because you now have two chances. You know one of them has a goat but the odds of a win are better.
    Tom

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    Default Re: Do you know the correct answer?

    I want an equation, I know it has to exists. else the answer is wrong!
    And 1 car 1 medium prize and 1 goat, not 2 goats!

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by ShorelineJohn View Post
    I want an equation, I know it has to exists. else the answer is wrong!
    And 1 car 1 medium prize and 1 goat, not 2 goats!
    The Harvard vid shows the equation, around 14 min mark. Worth watching.
    Tom

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by ShorelineJohn View Post
    I want an equation, I know it has to exists. else the answer is wrong!
    OK. Go here: https://medium.com/@judaikawa/the-mo...m-4415e50e233f. The article to which I've linked shows a different statistical equation solving the solution than that in the Harvard video (using the Law of Total Probability). The Harvard video is very good. It is also clear (I could follow it).

    After viewing the video and then reading the linked article and working through the Bayesian equation, actually play the game yourself. Do what JimmyW suggested, take an Ace and two Jokers out of a deck of playing cards. Enlist a friend to take the Monty Hall role. Play over and over for an hour. Bet a dollar to win on each play. Never switch. Be sure to return and tell us if your friend thanked you.
    .
    Last edited by Tom Montgomery; 06-16-2022 at 05:53 AM.
    "I'll tell you why [religion's] not a scam. In my opinion, all right? Tide goes in, tide goes out. Never a miscommunication. You can't explain that."Bill O'Reilly

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by lupussonic View Post
    What seems to be the consensus amongst some here is that the set of probabilities of first choice (1/3) is carried over somehow to the second set of probabilities in the second choice (1/2).

    Nick (and I) can not see that they are connected in anyway. How are the 2 sets of probabilities connected?

    To my mind it doesn't matter a damn which door I chose on first choice, or which door is the goat reveal, the second choice of 2 doors is completely unrelated to the first choice of 3 doors, yet some here say they are entangled some how?

    So what's the mechanism?
    You accept that the initial choice is a 1/3 change of getting the car.

    If you choose to switch your first choice for Monty's other door, then you reverse your choice - if you chose the car and switch, you get a goat, if you chose a goat and switch you get a car.

    Therefore, unless you can come up with a scenario where switching to Monty's unrevealed door does not reverse the outcome of your initial choice, switching doors must have a 2/3 chance of winning the car.
    'When I leave I don't know what I'm hoping to find. When I leave I don't know what I'm leaving behind...'

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    Default Re: Do you know the correct answer?

    .
    The offer is to switch your one door (with a 1/3 chance of having the car) for the aggregate of the remaining doors (which have a 2/3 chance of having the car). The odds the aggregate has a 2/3 chance of having the car do not change if one of the doors of the aggregate is revealed to not contain the car. You knew that already.
    "I'll tell you why [religion's] not a scam. In my opinion, all right? Tide goes in, tide goes out. Never a miscommunication. You can't explain that."Bill O'Reilly

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    Default Re: Do you know the correct answer?

    .
    It is a simple game to play with cards representing the doors. THEY WILL NOT PLAY.
    "I'll tell you why [religion's] not a scam. In my opinion, all right? Tide goes in, tide goes out. Never a miscommunication. You can't explain that."Bill O'Reilly

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    Default Re: Do you know the correct answer?

    the second choice of 2 doors is completely unrelated to the first choice of 3 doors, yet some here say they are entangled some how?
    After you pick, Monty has two doors. At least one of them will always hide a goat, and Monty know what's behind them. He away opens one with a goat, never the car. Thus, if you switch, you're getting the best of the two doors.

    Again, a proposition: We have two jokers and an ace. We shuffle and deal you one card, two for me. I'll always have a joker. I look at my cards, and show you a joker. We then turn over our remaining cards, and the ace wins. I'll play you for ten bucks a hand, for as long as you like. Try it yourself, it won't take long.
    "For a successful technology, reality must take precedence over public relations,
    for nature cannot be fooled."

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  19. #789
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    Default Re: Do you know the correct answer?

    .
    It is so simple. I am baffled why not one of the skeptics will play the game for themselves. Maybe because once observation contradicts one's hypothesis the hypothesis must be abandoned. I played the game 100 times. Had the total result been anywhere near 50/50 I would have admitted being in error.
    Last edited by Tom Montgomery; 06-16-2022 at 07:10 AM.
    "I'll tell you why [religion's] not a scam. In my opinion, all right? Tide goes in, tide goes out. Never a miscommunication. You can't explain that."Bill O'Reilly

  20. #790
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    Default Re: Do you know the correct answer?

    .
    I received a pm objecting that it is impossible to shuffle three cards and always have a random distribution.

    OK. Then play the game using three marbles rather than cards, say one black and two white. Play with a friend. Place the marbles in a bag, have your friend shake it well, and then you draw one marble from the bag. You must hold it in your clenched fist and not look at it. Your friend gets the bag and he may look inside to see what marbles he has. He may take out one white marble, show it to you, and then put it to one side. He then offers to switch the marble in his bag with the one in your hand. Turn down the offer. Reveal the marbles. Do this a number of times. Better?

    The other objection was that the game must be played hundreds of times to prove anything... and that life is too short. So probably not better.
    .
    Last edited by Tom Montgomery; 06-16-2022 at 07:36 AM.
    "I'll tell you why [religion's] not a scam. In my opinion, all right? Tide goes in, tide goes out. Never a miscommunication. You can't explain that."Bill O'Reilly

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by lupussonic View Post
    What seems to be the consensus amongst some here is that the set of probabilities of first choice (1/3) is carried over somehow to the second set of probabilities in the second choice (1/2).

    Nick (and I) can not see that they are connected in anyway. How are the 2 sets of probabilities connected?

    To my mind it doesn't matter a damn which door I chose on first choice, or which door is the goat reveal, the second choice of 2 doors is completely unrelated to the first choice of 3 doors, yet some here say they are entangled some how?

    So what's the mechanism?
    the mechanism is that all three doors are in the same game pool of three, and selecting the first of them removes it from that pool.
    it is a layer of probabilities in which the first pick affects the second pick in the game.
    even without agreeing on the exact number of probability, you cannot assert that the first pick has no bearing on the outcome of the game.

    we must accept that the player always has a 1 in 3 chance of winning the prize on their first guess.
    if Monty were playing against us, yes, his pick would only be from two doors.. but we could not say it is a 1 in 2 chance because that would not account for the 1 in 3 times the contestant might have won right from the start and Monty might, in fact, be holding two goats to choose from.

    at the very beginning of the game, there are three doors all with equal chance of hiding the prize (1/3 + 1/3 + 1/3)
    at the moment the player chooses their first door, the pool is split into two unequal parts: <- this is the big event and the mechanism that knocks the earth off its axis

    one player's door (with 1/3 chance of hiding the prize)
    and Monty's two doors (one door with 1/3 chance and the other door with equal 1/3 chance) Monty's probability 2/3 = 1/3 + 1/3

    there is never a 1 in 2 chance that either of Monty's doors has the prize, because that would not account for the first player's door which was loaded at the same time as Monty's
    the only 1 in 2 chance is whether Monty will open his left door or his right door, and that does not matter.

    even Nick, i think, would agree that at the point in the game when the first door was picked (but no doors were opened) the chances are 1/3 v. 2/3
    so what if, at this point in the game, Monty didn't offer his 'deal' but we all simply agreed to open all three doors at the exact same time ?
    what would happen is, player's door would win 1 in 3 times and Monty's two doors together as a whole would win 2 in 3 times.. there's no way around that

    when Monty decides to reveal a door and offer a deal before the final winner is decided, he does not alter the math in any way. he has just thrown in a bit of human psychology.

    lets think about it rationally..
    before the reveal, player's door has a 1/3 chance of winning the prize and Monty's two doors a total 2/3 chance
    we also know (without seeing) that one of Monty's doors must contain one goat. we know this because player's door can only hold one goat or the prize at one time, but there are two goats in the game.
    at this point (take a deep breath) Monty gingerly opens a door and reveals one of the game's two goats. this revelation only tells us specifically which door the goat is behind. but we are not actually interested in this because the game is not asking us to pick which of Monty's two doors has the prize. in the rules of the game, the final choice is from one of Monty's doors or the only one of the Player's door.
    the important part is, by opening one of his doors, Monty has reduced our choice from two down to the last one of his doors but not actually changed the math at all. that's because the total probability that his two doors contain the prize is still 2 in 3, even though we now only have one of his doors to choose from.

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    Default Re: Do you know the correct answer?

    .
    One's chances of choosing the one door of three containing a car is NEVER a 50% probability. It is ALWAYS a 33.33% probability. The skeptics would have us believe the way the game operates changes the odds off that initial choice. That is impossible.
    "I'll tell you why [religion's] not a scam. In my opinion, all right? Tide goes in, tide goes out. Never a miscommunication. You can't explain that."Bill O'Reilly

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    Default Re: Do you know the correct answer?

    If Monty initially opened one door to reveal a goat, then asked you to pick between the remaining two, the odds would indeed be 50-50. That is NOT the same thing, although the game is designed to fool you into thinking it is.
    "For a successful technology, reality must take precedence over public relations,
    for nature cannot be fooled."

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    Default Re: Do you know the correct answer?

    PM,

    I understand your position. I really do. For the first 14 pages of this thread I only lurked, because I intuitively thought like you - that it has to be 50-50 between two doors, and the opening of the goat door cannot affect the odds.

    But I also knew that this is wrong, not because I understood it, but because everyone, including unimpeachable sources, told me so. So I had to conclude that my logic was flawed, even if it seemed perfect to me.

    Only around page 14 (I actually only skimmed a few of the preceding posts) I took a walk with the dogs and tried to think about the problem with a different approach, and suddenly I understood that Monty has two doors. The one he opens is deliberately chosen to not reduce his odds of winning. Anyway, it doesn't matter how I saw the light. What matters is that you need to take fresh approaches to the problem until one of them gives you that aha! moment and you too see the light.

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by Tom Montgomery View Post
    .
    I received a pm objecting that it is impossible to shuffle three cards and always have a random distribution.

    OK. Then play the game using three marbles rather than cards, say one black and two white. Play with a friend. Place the marbles in a bag, have your friend shake it well, and then you draw one marble from the bag. You must hold it in your clenched fist and not look at it. Your friend gets the bag and he may look inside to see what marbles he has. He may take out one white marble, show it to you, and then put it to one side. He then offers to switch the marble in his bag with the one in your hand. Turn down the offer. Reveal the marbles. Do this a number of times. Better?

    The other objection was that the game must be played hundreds of times to prove anything... and that life is too short. So probably not better.
    .
    Ha! I like that. It's much better to waste ones life arguing about it for days on end!
    There's a lot of things they didn't tell me when I signed on with this outfit....

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by Keith Wilson View Post
    If Monty initially opened one door to reveal a goat, then asked you to pick between the remaining two, the odds would indeed be 50-50. That is NOT the same thing, although the game is designed to fool you into thinking it is.
    This is most easily seen with a deck of cards. If you are hoping for the ace of spades, you have 1/52 chance. If Monty randomly picks a card, his is 1/52. Now if you start turning over the other 50, 1 out of 26 times you won't find the ace of spades. So it is 50-50 between you and Monty. OTOH, if Monty goes through and specifically picks out the ACE of spades, leaves it turned over and then you go thought the other 50 cards, at that point Monty has a 51/52 chance of winning. Perhaps the thing that confuses people is stating the rule that Monty has to open a door with a goat. Instead look at it, he has to leave the car hidden.

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by Tom Montgomery View Post
    .
    I received a pm objecting that it is impossible to shuffle three cards and always have a random distribution.

    OK. Then play the game using three marbles rather than cards, say one black and two white. Play with a friend. Place the marbles in a bag, have your friend shake it well, and then you draw one marble from the bag. You must hold it in your clenched fist and not look at it. Your friend gets the bag and he may look inside to see what marbles he has. He may take out one white marble, show it to you, and then put it to one side. He then offers to switch the marble in his bag with the one in your hand. Turn down the offer. Reveal the marbles. Do this a number of times. Better?

    The other objection was that the game must be played hundreds of times to prove anything... and that life is too short. So probably not better.
    .
    Quote Originally Posted by George Jung View Post
    Ha! I like that. It's much better to waste ones life arguing about it for days on end!
    If played for $10 a hand, one will probably decide to quit giving away money long before 100s of times.

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    Default

    Quote Originally Posted by lupussonic View Post
    How can the probabilities of choosing between 2 different doors be 30% and 50%?


    The choice is between three doors. Two doors is just a stop enroute to the destination( end of game).


    Sent from my iPhone using Tapatalk Pro
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    Default Re: Do you know the correct answer?

    Exposing a card or door seems to be the mental hurdle. So just forget that and think about the odds with them hidden. One person has 2, one person has 1. Who would ever bet that the person with 1 has better odds of having the prize….?

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    Default Re: Do you know the correct answer?

    Quote Originally Posted by Decourcy View Post
    Exposing a card or door seems to be the mental hurdle. So just forget that and think about the odds with them hidden. One person has 2, one person has 1. Who would ever bet that the person with 1 has better odds of having the prize….?
    Apparently a couple people on the WBF.
    Tom

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    Default Re: Do you know the correct answer?

    Tom - you are a genius.

    I know... I know... you claim it was an accident.

    But that just makes you and Accidental Genius!!
    David G
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    Default Re: Do you know the correct answer?

    Quote Originally Posted by Decourcy View Post
    Exposing a card or door seems to be the mental hurdle. So just forget that and think about the odds with them hidden. One person has 2, one person has 1. Who would ever bet that the person with 1 has better odds of having the prize….?

    Maybe this situation has reached the "Trumpian phase," where the search for a cover up or face saving "out" is taking all the mental effort, and the truth of the problem has been abandoned...

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    Default Re: Do you know the correct answer?

    ^
    No. I believe those who believe the choice comes down to a 50/50 proposition are sincere in that belief. Any error must be in the assumptions behind their calculation.

    If it were me, and most people were telling me I was wrong, I would test my belief with a real-world test. That is in fact what happened with me. When I first encountered the problem I thought there was no advantage to switching. But upon investigating further I found an overwhelming number of sources providing logical explanation and statistical calculation that changed my mind.

    Nevertheless the solution to the problem certainly is counter-intuitive. Actually playing the game 100 times eliminated any remaining doubt I had regarding the advantage in switching. The critics claim that my results were due to being unable to randomly shuffle only three cards. I would play it again using three marbles. But I do not have marbles. Are they even commonly found for sale these days? In any event I suspect some other objection would discount the result of that form of the game as well.
    .
    Last edited by Tom Montgomery; 06-16-2022 at 05:11 PM.
    "I'll tell you why [religion's] not a scam. In my opinion, all right? Tide goes in, tide goes out. Never a miscommunication. You can't explain that."Bill O'Reilly

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    Default Re: Do you know the correct answer?

    I think the problem results from not realising that the host knows what is behind all three doors- once the goat is revealed, the contestant knows only two.

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    Default Re: Do you know the correct answer?

    I think the problem results from not realising that the host knows what is behind all three doors- once the goat is revealed, the contestant knows only two possibilities..

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