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Simple probability question

  1. Apr 10, 2014 #1
    1. The problem statement, all variables and given/known data

    My book says that
    P(C∩G) = P(G)P(C|G)

    but shouldn't it be

    P(C∩G)= P(C)P(C|G)

    ?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 10, 2014 #2

    jbunniii

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    No. To see why, consider the case where ##C## and ##G## are independent. Then we expect ##P(C|G) = P(C)## and ##P(C \cap G) = P(C)P(G)##. That is consistent with the formula from the book, not your proposed formula.
     
  4. Apr 10, 2014 #3

    Ray Vickson

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    The book is correct. The definitionof conditional probability is
    [tex] P(C|G) \equiv \frac{P(C \cap G)}{P(G)} \text{ if } P(G) \neq 0 [/tex]
    Sometimes, however, we are given ##P(C|G)## and ##P(G)##; in that case we can get ##P(C \cap G)## by 'reversing' the formula above.
     
    Last edited: Apr 10, 2014
  5. Apr 10, 2014 #4

    haruspex

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    Perhaps you are misreading the notation. P(C|G) means the probability of the event C given that event G occurs. Maybe you read it as the other way around?
     
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