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Math Help - Probability axioms problems

  1. #1
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    Probability axioms problems

    Problem #1

    If P(a) = .4 , P(b)= .5 and P(a ∩ b) = .3 find the following

    a). P (a U b) = .6 because P (a U b) = P(a) + P(b) - P(a ∩ b)

    b.) P(a' U b') = ?

    c.) P(a U b') = .3 because P (a U b) = P(a ∩ b') + P(a ∩ b)

    It's b.) that I'm not sure of since they're both compliments and I don't know what formula to use

    Also Problem #2 (separate probabilities from #1)

    If S = a U b , P(a) = .7, P(b) = .9, find

    P(a ∩ b) = ?

    and

    P( (a ∩ b)' ) = ?

    For these two problems I'm not really sure where to begin. Would S simply be 1 since P(S) is all elements of the sets and and b?
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  2. #2
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    Re: Probability axioms problems

    for (b)

    $$\left(a \bigcap b \right) = \left(a' \bigcup b' \right)$$

    for problem 2 S is just S. You don't know what it is so just call it S. Your answer will include it.

    use the above formula. use (1-p) for the probability of the complement

    for the 2nd one just do what you did in 1c with different numbers and the unknown S.
    Last edited by romsek; February 17th 2014 at 08:49 PM.
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  3. #3
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    Re: Probability axioms problems

    Thank you very much. As far as problem two is concerned, for the second part P( (a ∩ b)' ) is that the same as P(a' ∩ b') = P(a U b) ? because for P(a ∩ b) I got S. and for the 1st part of the problem I got S = .7 + .9 - P(a ∩ b) and thus P(a ∩ b)= 1.6 - S.
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  4. #4
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    Re: Probability axioms problems

    oh my mistake (c) was the union and you have the intersection here. Just apply demorgan's law

    $$(a \bigcap b)'=a' \bigcup b'$$

    and note that

    $$S'=a'\bigcap b'$$
    Thanks from crownvicman
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  5. #5
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    Re: Probability axioms problems

    Hello, crownvicman!

    Did you draw a Venn diagram?


    \text{If }P(A) = 0.4,\;P(B)= 0.5,\;P(A\cap B) = 0.3,\;\text{ find the following:}

    . . (a)\;P(A \cup B) \qquad (b)\;P(A' \cup B') \qquad (c)\;P(A \cup B')
    Code:
        *-------------------------------------*
        |                                     |
        |  U        * * *       * * *         |
        |       *   A       *       B   *     |
        |     *           *   *           *   |
        |    *           *     *           *  |
        |                                     |
        |   *           *       *           * |
        |   *    0.1    *  0.3  *    0.2    * |
        |   *           *       *           * |
        |                                     |
        |    *           *     *           *  |
        |     *           *   *           *   |
        |       *           *           *     |
        |  0.4      * * *       * * *         |
        |                                     |
        *-------------------------------------*
    Now you can answer the questions.





    \text{If }S \,=\,A\cup B,\; P(A) = 0.7,\;P(B) = 0.9,

    . . \text{Find: }(1)\;P(A\cap B)\;\;\;(2)\;P(A\cap B)'

    I assume that S is the universal set.

    Then the diagram looks like this:

    Code:
                  * * *       * * *
              *   A       *       B   *
            *           *   *           *
           *           *     *           *
    
          *            *      *           *
          *    0.1    *  0.6  *    0.3    *
          *           *       *           *
    
           *           *     *           *
            *           *   *           *
              *           *           *
                  * * *       * * *
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