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Math Help - This conditional probability problem makes no sense to me intuitively

  1. #1
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    This conditional probability problem makes no sense to me intuitively

    "The king comes from a family of 2 children. What is the probability that the other child is his sister?"

    I know if I work through this problem using conditional probability concepts, the answer is 2/3.

    However, could someone explain why this is the answer?

    I don't see why this question is any different from the following:

    "A coin is flipped 2 times. The first flip lands on heads. What is the probability that the second flip lands on tails?"

    Yet the answer to the second question seems to obviously be 1/2.
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  2. #2
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    Quote Originally Posted by paulrb View Post
    "The king comes from a family of 2 children. What is the probability that the other child is his sister?"

    I know if I work through this problem using conditional probability concepts, the answer is 2/3.

    However, could someone explain why this is the answer?

    I don't see why this question is any different from the following:

    "A coin is flipped 2 times. The first flip lands on heads. What is the probability that the second flip lands on tails?"

    Yet the answer to the second question seems to obviously be 1/2.
    King question: The king is a boy. The sample space for the King question is therefore BG, GB, BB where BG means the king has an older sister and GB means that the king has a younger sister. Two of the three outcomes are favourable to the king having a sister.


    Coin question: "A coin is flipped 2 times. The first flip lands on heads. What is the probability that the second flip lands on tails?"

    The sample space is HT, HH. One of the two outcomes are favourable to the second toss being a tail.


    Alternate coin question (King question in disgiuse): "A coin is flipped 2 times. One of the flips lands on heads. What is the probability that the other flip landed on tails?"

    The sample space is HT, HH, TH. Two of the three outcmes are favourable to the other flip being a tail ....
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  3. #3
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    Thank you, that helps, but I don't understand why I'm incorrect using this reasoning:

    We know the king has a sibling. The sibling may be:
    1) An older sister
    2) A younger sister
    3) An older brother
    4) A younger brother

    2 of those possibilities involve the other sibling being a sister, and 2 of the possibilities involve the other sibling being a brother. Assuming each possibility is equally likely, there is a 2/4 chance the sibling is a brother and a 2/4 chance the sibling is a sister. Or 1/2 for each.
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    Quote Originally Posted by paulrb View Post
    "The king comes from a family of 2 children. What is the probability that the other child is his sister?"

    I know if I work through this problem using conditional probability concepts, the answer is 2/3.

    However, could someone explain why this is the answer?

    I don't see why this question is any different from the following:

    "A coin is flipped 2 times. The first flip lands on heads. What is the probability that the second flip lands on tails?"

    Yet the answer to the second question seems to obviously be 1/2.
    Hi Paulrb,

    I'm going to disagree with your first answer. The probability that the other child is a girl is 1/2, for the reasons stated in your analysis of the coin flip problem.

    Don't confuse your king-sibling problem with the classic problem where we are told that at least one child is a boy and we are asked to find the probability that both children are boys. In your problem we have more information: we know the king is male and we are asked about the probability that his sibling is female (or male, it makes no difference.) This is different than being told that at least one child is a boy.

    Maybe this will be clearer if we employ some symbols. Let X_i = 1 if the ith child is a boy, zero otherwise. In the king-sibling problem we are told X_1 = 1 and we are asked to find P(X_2 = 1). In the "at least one boy" version we are told X_1 + X_2 \geq 1 and we are asked to find P(X_1 = 1 \text{ and } X_2 =1). These are two different problems.

    It's very difficult to come up with a real-world situation where all you know is that at least one child is a boy. Usually you have seen one child, and that child was a boy; that's different. That is one of the reasons why the "at least one..." problem gives people so much trouble.
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  5. #5
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    I may be beating that ole dead horse, but here goes.
    GAME: I toss two coins out of your sight. Then cover the outcome, only you can remove the covering. You then declare if both coins show the same side or if both coins show different sides. You then remove the cover to see if you win.

    Now you absolutely know the possible outcomes.
    \begin{array}{cc}   H & H  \\   H & T  \\   T & H  \\   T & T  \\ \end{array}
    So you can easily see that either choice is equally likely. So you choose either.

    But now I condition the game by telling you that one coin is heads.
    Now you also absolutely know the possible outcomes.
    \begin{array}{cc}   H & H  \\   H & T  \\   T & H  \\ \end{array}
    Would you now choose the option that both are the same?
    Of course not! Choosing the option that “they are different” is clearly best.

    Conditional probabilities has the effect of narrowing the possible outcomes.
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    Is this problem different though? The siblings aren't identical objects; they're distinct people who were born at different times. It may be more like flipping a penny and a quarter.

    We are told one of the coins lands on heads, without being told whether that coin is a penny or a quarter. Therefore, the other coin may be:
    1) A quarter that landed on heads
    2) A quarter that landed on tails
    3) A nickel that landed on heads
    4) A nickel that landed on tails.

    Meaning, a 1/2 probability of landing on heads if each of the above possibilities is equally likely. We cannot rule out any of the above possibilities because we were only given half the information required to do so. We need to know the coin's denomination *as well as* whether it landed on heads or tails, but we are not told the coin's denomination, only that it landed on heads. Similarly, we were not told if the king was born first or second.

    To awkward - I said 2/3 because that's what the solutions manual for my book says. I typed the problem verbatim. Also, if you search Google for this problem, other people have worked solutions for it and arrived at 2/3. To be honest, I am still not fully convinced that is the correct answer, but then again I don't think so many people would be incorrect.

    EDIT: Shortly after typing this I think I figured it out. Since we *are* told one of the coins landed on heads, there is a 1/2 chance the quarter may NOT land on tails, since it landed on heads. There is also a 1/2 chance the penny may NOT land on tails, since it landed heads. Therefore, each of those possibilities should only count half.

    1 * quarter lands on heads (there is no way that possibility can be ruled out)
    .5 * quarter lands on tails (there is a 1/2 chance that possibility can be ruled out)
    1 * penny lands on heads (there is no way that possibility can be ruled out)
    .5 * penny lands on tails (there is a 1/2 chance that possibility can be ruled out)

    So with the "weighted" possibilities, there's a 2/3 chance the other coin lands on heads and a 1/3 chance it lands on tails. The same logic applies to my problem.

    Except...wait, did I get something backwards? In the king problem there was a 2/3 chance the other sibling was a girl (opposite sex of the other) but in this case there's a 2/3 chance the other coin lands on heads (same side as the other)
    Last edited by paulrb; February 10th 2009 at 04:32 PM.
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  7. #7
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    Yes, having thought the problem over, I can see that the "at least one boy" interpretation is probably (no pun intended) the intent of the author of the problem. In that case the probability that the other child is a girl is 2/3, as others have pointed out.

    However, I think there are other interpretations possible, and quite likely in the real world. When the problem says "the king comes from a family of two children", what does that mean? If it means that someone saw one of the two children and that one happened to be male, then the probability that the other child is female is 1/2. But if it means no more and no less than that at least one of the children is male, then the probability that the other child is female is 2/3. Altogether, I think the problem statement is sufficiently ambiguous to allow for either possibility.
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  8. #8
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    Quote Originally Posted by paulrb View Post
    Is this problem different though? The siblings aren't identical objects; they're distinct people who were born at different times. It may be more like flipping a penny and a quarter.
    EDIT: Shortly after typing this I think I figured it out. Since we *are* told one of the coins landed on heads, there is a 1/2 chance the quarter may NOT land on tails, since it landed on heads. There is also a 1/2 chance the penny may NOT land on tails, since it landed heads. Therefore, each of those possibilities should only count half.
    No, I think you are confusing yourself.
    The point that you are missing is this: even though siblings are distinct, they are still animals and as such they are male or female. Likewise, even though coins are distinct, they still have only two sides, heads or tails.
    It is binary, if we know that one of two outcomes is a 1, then there are just three possible outcomes.
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    Ok... I understand what you're saying. Still, what I said from my previous post is also true:

    Do you agree with this:
    We know the king has a sibling. The sibling may be:
    1) An older sister
    2) A younger sister
    3) An older brother
    4) A younger brother
    Even if this is a more complicated way to do the problem, I would like to do it this way to convince myself that the result is the same.

    Is there a way to show that these 4 possibilities are *not* equally likely, and that they are skewed so that there is a 2/3 chance the other sibling is a girl?

    Edit: Perhaps this is the reason why the problem specifically stated "the king?" Most people know that it's the firstborn who becomes king, not the second born. Therefore, the probability that the sibling is an older brother is 0. Then there are 3 possibilities, giving a 2/3 chance that the other sibling is his sister.

    Maybe that's a coincidence, though. Still, it makes all the difference.
    Last edited by paulrb; February 10th 2009 at 05:36 PM.
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