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Thread: Disjoint/Independent Events

  1. #1
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    Exclamation Disjoint/Independent Events

    Let E be an experiment with sample space S. Let A and B be events in S, where A and B both have positive probability.
    i) Show that if A and B are disjoint, then they are NOT independent.
    ii) Show that if A and B are independent, then they are NOT disjoint.

    I am getting more familiar with basic probability but where would I start for this? Also, what makes two events disjoint/independent?
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  2. #2
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    Re: Disjoint/Independent Events

    Apparently, you don't know the definitions of disjoint and independent. Event A and B are disjoint iff $A\cap B=\emptyset$ and A and B are independent iff $P(A\cap B)=P(A)P(B)$.
    Now suppose both $P(A)>0$ and $P(B)>0$.
    1. Suppose A and B are disjoint. Can A and B be independent? Remember $P(\emptyset)=0$.
    2. Suppose A and B are independent. Can A and B be disjoint? Remember the product of two positive reals is positive.
    Thanks from HallsofIvy and azollner95
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    Re: Disjoint/Independent Events

    Thank you for explaining that. This now makes sense, I am just curious how to actually word this in a concise format.
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    Re: Disjoint/Independent Events

    Quote Originally Posted by azollner95 View Post
    I am just curious how to actually word this in a concise format.
    Assuming that you know logic, let $D$ be the statement that "two events with positive probability are disjoint"; let $I$ be the statement that "two events with positive probability are independent".
    The part A) says If D then not I. In symbols $D \Rightarrow \;\neg I$ That is equivalent to $\neg D\vee\neg I$

    The part B) says If I then not D. In symbols $I \Rightarrow \;\neg D$ That is equivalent to $\neg I\vee\neg D$

    Look at this truth-table.

    Can we say "That two events with positive probability are not independent OR not disjoint."?
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