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  1. #101
    Protocol Droid Athenian200's Avatar
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    I'm not sure what the answer to the card one was, but I somehow knew "B" wasn't relevant, and that "A" was reasonable.

    Because if I turned over "A," and found an odd number, the statement would be false.

    I also knew that if I turned over "7," and found a vowel, the statement would be false.

    However, my error was in assuming that if I turned over "4," and found a consonant, then the statement would be false. (Why did I think this? I guess I assumed that the statement could be reversed. "If a card has an even number on one side, then it must have a vowel on the other," and assumed that if the even number didn't have a vowel, then the statement was false. What kind of error is this?)

    I already knew that "B" couldn't prove/disprove anything about the statement, because it was about vowels, and couldn't fall into the condition stated.

    I was most positive of "A" and felt a bit shaky on the numbers, but not sure why.

    So, I thought that you would have to turn over every card except "B" in order to know for certain whether the statement was true or false, but that my best bet for invalidating it would lie with "A."

    I failed to eliminate "4," in other words.

  2. #102
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by ygolo View Post
    Another way to get a probablity of 1/2 is to simply say the producers will once again randomly place the desired prize behid the remaining doors after one door is opened.

    Also....

    if we were to modify the problem so that you are allowed three options, stay with your door or switch to either of the two other doors. Let's say now, due to theatrics by the host, you were able to accidentally glimpse behind a door you didn't pick to see a booby prize, instead of what you wanted. Obviously, you wouldn't pick the door with the booby prize, but would you keep your door, or switch? Does it make a difference?
    Yes those are also ways to produce a 1/2 probability. My point was actually to produce the scenario which I believe is the most likely in reality. That is even if the scenario is not a "50-50 chance" the first time the game is played, it will eventually become that way...for economic reasons rather than for purely mathematical ones. I think it is likely that Monty Hall will actually open all of the doors sometimes (and if you've seen "Let's Make a Deal", you see that it actually happens). And he uses this opening all of the doors mechanism to produce the "50-50 chance" of a guess on the occasions that he does eliminate one of the doors.

    My point is basically this. A person who is not mathematically trained will intuitively say that the probability is 1/2. A person who is properly mathematically trained will say the probability is 2/3. However the math problem is actually an artificial construct. In reality the probability is closer to 1/2. Logic is more useful when applied to abstract problems, but intuition is actually more acurate when it comes to what actually happens in reality. Sometimes learning can "untrain" what would normally be good common sense.
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  3. #103
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    I prefer this problem to the "Monty Hall" problem, because I don't think you have to make as many assumptions to solve it. (The problem below assumes that when a woman gives birth there is an equal chance of the child being either a boy or a girl.)

    My neighbor has two children. One of them is a boy. What is the probability that the other child is a girl?
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  4. #104
    Protocol Droid Athenian200's Avatar
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    Quote Originally Posted by The_Liquid_Laser View Post
    My point is basically this. A person who is not mathematically trained will intuitively say that the probability is 1/2. A person who is properly mathematically trained will say the probability is 2/3. However the math problem is actually an artificial construct. In reality the probability is closer to 1/2. Logic is more useful when applied to abstract problems, but intuition is actually more acurate when it comes to what actually happens in reality. Sometimes learning can "untrain" what would normally be good common sense.
    Thanks, you explained my main objection to the answer better than I could have, I didn't think to bring in outside elements. But I now agree that the answer in the specifically given scenario, repeated several times, would be 2/3, but it was like pulling teeth to understand how that could be right.

    I really think only an IxTP could just accept logic like that without an inner struggle (that's their primary strength). I have to understand some reason why something is true in a way that makes sense to me, otherwise it simply isn't, regardless of what my data tells me. That's why I didn't understand it until I looked at the situation from the host's perspective, and realized that in 2/3 of the cases, he would be forced to open the unchosen goat door, and leave the remaining one to be the car. He would only be free to open either of the alternate doors in the 1/3 of cases where you indeed chose the car.

    But this whole scenario is in sort of a logical vacuum. To be honest, I can't fathom why I've spent so much time trying to solve a completely meaningless problem with no particular motivation to do so. I don't understand why I would care what the answer was, it's just a riddle, and I feel kind of silly now that I've let myself be so consumed with this. Curiosity killed the cat, you know?

    I guess it's that darn 5 wing of mine.

  5. #105
    Protocol Droid Athenian200's Avatar
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    Quote Originally Posted by The_Liquid_Laser View Post
    I prefer this problem to the "Monty Hall" problem, because I don't think you have to make as many assumptions to solve it. (The problem below assumes that when a woman gives birth there is an equal chance of the child being either a boy or a girl.)

    My neighbor has two children. One of them is a boy. What is the probability that the other child is a girl?
    I... think it would still be 1/2. I'm probably wrong again, aren't I? You're all going to find some crazy mathematical esoteric stuff that will somehow make sense logically, and prove that I'm wrong.

    My reasoning, however is this. The gender of the first child was determined by a 1/2 chance as an independent event, as was the gender of the second child. I'm sure the correct answer, whatever it is, finds a way to explain the influence of the one child's gender on the gender of the other, but I just can't see it.

  6. #106
    Feline Member kelric's Avatar
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    I sat there for a while on the same thought path that Athenian is working with... but I once I got it I think I understand... think of it like this:

    Step 1: You select one of three doors - say you pick door A.

    Now, after step one, I think we can all agree that there is a 33% chance that the car is behind door A, and a 66% chance that it is behind *either* door B or C. If you were to be asked at this point, "Do you want door A, which you picked, or both doors B and C?" everyone would probably agree that B+C is the better bet. One other thing we know is that there is a 100% chance that there is a goat behind *either* B or C - absolutely, no doubt about it - there might be goats behind both, but there is *definitely* at least one.

    The trick is that being shown the goat behind one of the two doors you didn't choose is completely irrelevant. Right now, after your first choice, there is a 100% chance that there is a goat behind at least one of B and C. You ignore Monte as he makes a big show about revealing the goat you knew was there.

    Step 2. Now, Monte asks you "Do you want to switch?" What he's really asking you is "Do you want what you have (33%) or what you didn't choose (66%)?" Since you knew beforehand that there is a 100% chance that one of your non-chosen doors was hiding a goat, you gained no additional information by being shown that there is a goat behind one of the two. It's *still* 33% vs. 66%.

    It certainly is counterintuitive, though. I was trying to think of why it's so difficult (for me, anyway) to grasp. The only thing I can think of is that it's natural to assume that we gained information when Monte reveals a goat, and it kicks us into thinking that we need to reevaluate based on new information - when in fact, we haven't gained any information pertinent to the problem.

  7. #107

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    Quote Originally Posted by athenian200 View Post
    I'm not sure what the answer to the card one was, but I somehow knew "B" wasn't relevant, and that "A" was reasonable.

    Because if I turned over "A," and found an odd number, the statement would be false.

    I also knew that if I turned over "7," and found a vowel, the statement would be false.

    However, my error was in assuming that if I turned over "4," and found a consonant, then the statement would be false. (Why did I think this? I guess I assumed that the statement could be reversed. "If a card has an even number on one side, then it must have a vowel on the other," and assumed that if the even number didn't have a vowel, then the statement was false. What kind of error is this?)

    I already knew that "B" couldn't prove/disprove anything about the statement, because it was about vowels, and couldn't fall into the condition stated.

    I was most positive of "A" and felt a bit shaky on the numbers, but not sure why.

    So, I thought that you would have to turn over every card except "B" in order to know for certain whether the statement was true or false, but that my best bet for invalidating it would lie with "A."

    I failed to eliminate "4," in other words.
    Unfortunately, the word "if" in English often means "if and only if", that's why I actually give the statement as something like "All vowels will have even numbers on the other side."

    Incidentally, if you interpreted the statement to also mean "If a card has an even number on one side, then it must have a vowel on the other," you would need to flip over "B" to make sure it is not an even number on the other side. So it is still logically incorrect to pick the three cards you did based on the interpretation of "A card has an even number on one side, if and only if it has a vowel on the other."

    Why does intuition go wrong here?

    I think intuition has its place in situations people are familiar with (like checking for legal drinking age).

    But it is logic (combined with imagination) that gets us to guess things like the existence of anti-particles, and allows us to deal with the wave-particle duality, or even that F=ma (I mean the second derivative in there? Intuitive, really?)

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  8. #108
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by athenian200 View Post
    I'm probably wrong again, aren't I? You're all going to find some crazy mathematical esoteric stuff that will somehow make sense logically, and prove that I'm wrong.
    Heh, yes I think this is a case where experience with probability is of great help. In particular you will find that additional information is rarely trivial when it comes to determining probability. (Yes sometimes it is thrown in there to confuse, but usually you will find that it does matter.) This basically has to do with the idea of conditional probability. "If we know this extra information then what is the probability?"


    But this whole scenario is in sort of a logical vacuum. To be honest, I can't fathom why I've spent so much time trying to solve a completely meaningless problem with no particular motivation to do so. I don't understand why I would care what the answer was, it's just a riddle, and I feel kind of silly now that I've let myself be so consumed with this. Curiosity killed the cat, you know?
    Well many people consider solving puzzles to be fun. On top of that solving math/logic puzzles, I believe, is the mental equivalent of lifting weights. When people lift weights they don't directly do anything useful. I mean they could be using those muscles to build a house or clean up or change a tire or... in other words something obviously productive. But lifting weights is a very efficient way to work out muscles and build muscle mass. In the same way I believe puzzles like these can give the brain a good workout. At the same time though I think it is good to remember that these puzzles are usually artificial constructs and may not reflect reality.
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  9. #109
    Oberon
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    Quote Originally Posted by athenian200 View Post
    I... think it would still be 1/2. I'm probably wrong again, aren't I? You're all going to find some crazy mathematical esoteric stuff that will somehow make sense logically, and prove that I'm wrong.

    My reasoning, however is this. The gender of the first child was determined by a 1/2 chance as an independent event, as was the gender of the second child. I'm sure the correct answer, whatever it is, finds a way to explain the influence of the one child's gender on the gender of the other, but I just can't see it.
    No, you're right. It's about 1 chance in 2. (Actually the odds of it being a girl are very slightly higher, as a few more girls are born each year than boys; but that's a quibble.)

    One of my math teachers in high school used to like to stump people with the following problem:

    "You flip a coin ten times in a row (without cheating), and it comes up heads each time. What's the probability that, if you flip it the eleventh time, it'll come up heads again?"

    Many students calculate the odds of getting 11 heads in a row, which are quite remote; but in probability, events that already occurred are red herrings and have no bearing on what will happen in the future. So the answer is 50%, or 1 in 2.

  10. #110
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by oberon View Post
    No, you're right. It's about 1 chance in 2. (Actually the odds of it being a girl are very slightly higher, as a few more girls are born each year than boys; but that's a quibble.)

    One of my math teachers in high school used to like to stump people with the following problem:

    "You flip a coin ten times in a row (without cheating), and it comes up heads each time. What's the probability that, if you flip it the eleventh time, it'll come up heads again?"

    Many students calculate the odds of getting 11 heads in a row, which are quite remote; but in probability, events that already occurred are red herrings and have no bearing on what will happen in the future. So the answer is 50%, or 1 in 2.
    No the answer is not 1/2. I can assure you that the problem I'm presenting is not equivalent to the one you are describing with coin flips.
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