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Math Help - conditional mini max

  1. #1
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    Exclamation conditional mini max

    what does minimum and maximum values of a conditional probability P(A/B) mean?
    When P(A)=0.3, P(A ∩ B)=0.1
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  2. #2
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    Hello, amul28!

    An interesting problem . . . It takes a bit of Thinking.


    What does minimum and maximum values of a conditional probability P(A|B) mean?
    When P(A)=0.3,\; P(A\cap B)=0.1

    Bayes' Theorem: . P(A|B) \:=\:\dfrac{P(A \cap B)}{P(B)}

    And we have: . P(A|B) \;=\;\dfrac{0.1}{P(B)} .[1]


    What is the range of values for P(B)?


    The Venn diagram looks like this:


    Code:
    
          * - - - - - - - - - *
          |                   |
          |   *-------*       |
          |   | A     |       |
          |   |0.2*---+---*   |
          |   |   |   |   |   |
          |   |   |0.1|   |   |
          |   *---+---*   |   |
          |       |     B |   |
          |       *-------*   |
          |                   |
          * - - - - - - - - - *

    We are told that: . P(A \cap B) \:=\:0.1
    That goes in the region common to circles \,A and \,B.

    We are told that: . P(A) = 0.3
    The 0.3 goes in the entire \,A-circle.
    That leaves 0.2 for the (A \cap B') region,
    . . the region that is in \,A, but not in \,B.

    What values can go in the region (B \cap A') ?
    . . The value can range from 0.0 to 0.7.
    Hence, P(B) can range from 0.1 to 0.8.


    Substitute into [1].

    \text{If }P(B) = 0.8,\text{ then: }\;P(A|B) \:=\:\dfrac{0.1}{0.8} \:=\:0.125\;\text{ minimum}

    \text{If }P(B) = 0.1,\text{ then: }\;P(A|B) \:=\:\dfrac{0.1}{0.1} \:=\:1.0 \quad\text{ maximum}

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  3. #3
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    thank u.

    even i found the same way but got messed up near 0.8 now its done.
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