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Math Help - Conditional Probabilty

  1. #1
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    Conditional Probabilty

    Suppsoe it takes at least 10 votes of guilty from a 12-member jury to convice a defendant. Suppose that the probability an individual juror votes a guilty person innocent is 0.2, whereas the probability a juror votes an innocent person guilty is 0.1, and assume jurors reach their decisions independently.

    (a) What is the distribution of the number of votes of guilty if the defendant is innocent, and if they are guilty?
    (b) Find the probability a defendant is convicted given that he is guilty, and find the probability a defendant is convicted given that he is innocent.
    (c) If 65% of all defendants are guilty, what is the probability the jury reaches the correct decision on a randomly selected defendant?
    (d) Again, if 65% of all defendants are guilty, what is the probability that a convicted defendant is actually innocent?

    Does anyone know how to do this? I don't understand part (a) and think i may need that for the other parts?
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  2. #2
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    Quote Originally Posted by Revilo View Post
    (b) Find the probability a defendant is convicted given that he is guilty,
    He can be convicted guilty or he can be convited innocent. (We are assuming he is really guilty).

    Each person has a .8 chance of convicting him guilty. So the probability that he is convicted guilty is \sum_{n=10}^{12}C_{12,n}(.8)^n(.2)^{12-n}.

    Each person has a .2 chance of convicting him innocent. So the probability that he is convicted innocent is \sum_{n=10}^{12}C_{12,n}(.2)^n(.8)^{12-n}.

    Now sum up these two probabilities to get your answer.
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    Quote Originally Posted by Revilo View Post
    (c) If 65% of all defendants are guilty, what is the probability the jury reaches the correct decision on a randomly selected defendant?
    Let C be the event that they make a correct decision. Let I be the event that he is innocent and G be that he is guilty (so G=I^C).

    Then,
    P(C) = P(C|I)P(I)+P(G|I^C)P(I^C) = P(C|I)P(I)+P(C|G)P(G)

    In this case P(I)=.35 and P(G) = .65.
    Thus, P(C) = (.35)P(C|I)+(.65)P(C|G).

    Now P(C|I) is the probability that the jury reaches a correct decision given that he is innocent. So we need 10 or 11 or 12 to convict him innocent. The probability that an individual member says innocent is .9. Thus, the overal probability that we seek is \sum_{n=10}^{12}  C_{12,n}(.9)^n (.1)^{12-n}. Do the same with the other one.
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    Thanks for the replys. I can follow your reasoning, but i don't understand what the C_{12,n} stands for in the following equations:

    Quote Originally Posted by ThePerfectHacker View Post
    \sum_{n=10}^{12}C_{12,n}(.8)^n(.2)^{12-n}.
    \sum_{n=10}^{12}C_{12,n}(.2)^n(.8)^{12-n}.
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    Quote Originally Posted by Revilo View Post
    Thanks for the replys. I can follow your reasoning, but i don't understand what the C_{12,n} stands for in the following equations:
    It is a binomial coefficient:

    C_{12.n} = \frac{12!}{(12-n)!n!}

    Though having both parameters as subscripts is not standard notation.

    RonL
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    What he means:

    there are 12 jurors. Of these 12 jurors, at least 10 are going to vote GUILTY if he is to be convicted guilty. But it could be any permutation of the 12 jurors. So we do {12 \choose 10}+{12 \choose 11} + {12 \choose 12}=\sum_{n=10}^{12}{12 \choose n}

    But I have no idea what is this question asking about:

    (a) What is the distribution of the number of votes of guilty if the defendant is innocent, and if they are guilty?
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  7. #7
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    Quote Originally Posted by ThePerfectHacker View Post

    Now P(C|I) is the probability that the jury reaches a correct decision given that he is innocent. So we need 10 or 11 or 12 to convict him innocent. The probability that an individual member says innocent is .9. Thus, the overal probability that we seek is \sum_{n=10}^{12} C_{12,n}(.9)^n (.1)^{12-n}. Do the same with the other one.

    Can you convict someone "innocent"? I thought we can only convict someone "guilty", by voting at least 10, and anything less than that would be innocent. So 0 to 9 votes of GUILTY corresponds with 12 to 3 votes of INNOCENT.

    So, isn't the answer:
    \sum_{n=3}^{12} C_{12,n}(.9)^n (.1)^{12-n}
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