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Math Help - Help Understanding Conditional Probability

  1. #1
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    Help Understanding Conditional Probability

    I am in a finite mathematics class and have been given the following Pr[A|B] = 3/4 Pr[B|A] = 3/8 and Pr[A u B] = 1
    I have been asked to find Pr[A] Pr[B] and Pr[A n B]
    the book has provided the formula:

    Pr[A|B] = Pr[A n B] Pr[B|A] = Pr[B n A]
    Pr[B] Pr[A]

    I am having difficulty understanding where to begin with this. Thanks.
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  2. #2
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    Note that Pr[A u B] = Pr[A] + Pr[B] - Pr[A n B].
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  3. #3
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    Quote Originally Posted by batman View Post
    Note that Pr[A u B] = Pr[A] + Pr[b] - Pr[A n B].
    Batman,
    Thanks for the reply. Given Pr[A|B] = 3/4 Pr[B|A] = 3/8 and Pr[A u B] = 1. How do I arrive at Pr[A] or Pr[B]?
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  4. #4
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    Quote Originally Posted by mlynch18 View Post
    I am in a finite mathematics class and have been given the following Pr[A|B] = 3/4 Pr[B|A] = 3/8 and Pr[A u B] = 1
    I have been asked to find Pr[A] Pr[b] and Pr[A n B]
    the book has provided the formula:

    Pr[A|B] = Pr[A n B] Pr[B|A] = Pr[B n A]
    Pr[b] Pr[A]

    I am having difficulty understanding where to begin with this. Thanks.
    \Pr(A | B) ~ \Pr(B) = \Pr(A \cap B) \Rightarrow \frac{3}{4} ~ \Pr(B) = \Pr(A \cap B) .... (1)

    \Pr(B | A) ~ \Pr(A) = \Pr(A \cap B) \Rightarrow \frac{3}{8} ~ \Pr(A) = \Pr(A \cap B) .... (2)

    Equate equations (1) and (2):  \frac{3}{4} ~ \Pr(B) = \frac{3}{8} ~ \Pr(A) \Rightarrow \Pr(A) = 2 \Pr(B) .... (3)

    \Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)

    Substitute from equations (3) and (1):

    \Rightarrow 1 = 2 \Pr(B) + \Pr(B) - \frac{3}{4} \Pr(B).

    Solve for Pr(B). Hence solve for Pr(A) and \Pr(A \cap B).
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