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Thread: Independent Events

  1. #1
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    Independent Events




    Hello,

    Can anyone assist with this tricky problem ?

    A and B are two independent events such that:

    P(A)+P(B)=t, where t is a number in the close set [0,2].

    What is the maximum possible value of P(A and B) (intersection).

    How do I even start ?

    Thanks !

  2. #2
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    Re: Independent Events


    Let's call x = P(A) and y = P(B).

    Since A and B are independent we know that P(A \cap B) = P(A)P(B) = xy

    Our goal is to maximize xy with the constraint that x + y = t (in other words... y = t-x).

    This reduces down to a simple calculus problem (or just knowing where the maximal value of a quadratic function occurs) once you make the correct substitution.
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