Results 1 to 3 of 3

Math Help - [SOLVED] Mutually independent

  1. #1
    Member
    Joined
    Jul 2008
    Posts
    119

    [SOLVED] Mutually independent

    Problem:
    Let the three mutually independent events C_{1},C_{2},C_{3}, be such that P(C_{1}) = P(C_{2}) = P(C_{3}) = \frac{1}{4}.

    Find  P[(C_{1}^c \cap C_{2}^c) \cup C_{3}].


    Attempt:

    By the definition of mutually independent events,

     P(C_{1} \cap C_{2} \cap C_{3}) = P(C_1) P(C_2) P(C_3).

    Since they are mutually independent events, they are also pairwise e.g.

    P(C_1 \cap C_3) = P(C_1) P(C_3)

    So, by the distribution property i.e.  A \cup (B \cap C) = (A \cup B) \cap (A \cup C) , then

     P[(C_{1}^c \cap C_{2}^c) \cup C_{3}] = P[(C_{1}^c \cup C_3 ) \cap (C_2^c \cup C_3)]

    Then by DeMorgan's Laws,

     = P [(C_1 \cap C_{3}^c)^c \cap (C_2 \cap C_{3}^c)^c)]

    Since C1, C2, and C3 are mut. ind., then

     = P((C_1 \cap C_{3}^c)^c P((C_2 \cap C_{c}^c)

    But from here I do not know how to continue.

    Thank you for your time.
    Last edited by Paperwings; February 2nd 2009 at 12:28 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    Recall the that complements of independent events are also independent.
    P\left[ {\left( {A^c  \cap B^c } \right) \cup C} \right] = P\left( {A^c  \cap B^c } \right)  + P(C) - P\left[ {\left( {A^c  \cap B^c } \right) \cap C} \right] = P\left( {A^c } \right)P\left( {B^c } \right) + P(C) - P\left( {A^c } \right)P\left( {B^c } \right)P\left( C \right)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2008
    Posts
    119
    Ah, yes. I completely forgot about that. Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Events A,B (independent, mutually exclusive), Help!
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: February 23rd 2011, 04:07 PM
  2. [SOLVED] Prob(a or b or c ... or z) NOT mutually exclusive
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: March 22nd 2010, 09:01 AM
  3. [SOLVED] Mutually Exclusive Events
    Posted in the Statistics Forum
    Replies: 1
    Last Post: April 6th 2009, 02:17 AM
  4. [SOLVED] Mutually Indpendent Question
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: September 7th 2008, 08:56 AM
  5. please help.mutually independent problem
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: August 28th 2008, 08:57 PM

Search Tags


/mathhelpforum @mathhelpforum