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Math Help - simplified prove of bonferroni inequality

  1. #1
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    simplified prove of bonferroni inequality

    If A and B are 2 events, prove that P( A ∩ B ) ≥ 1 - P(A') -P(B') and hence P( A ∩ B ∩ C ) ≥ 1 - P(A') -P(B') - P(C').

    thanks! I haven learn about bonferroni inequality..so i dont know how to use it though
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    If A and B are 2 events, prove that P( A ∩ B ) ≥ 1 - P(A') -P(B') and hence P( A ∩ B ∩ C ) ≥ 1 - P(A') -P(B') - P(C').
    Note that P(A\cup B)\le 1 so -P(A\cup B)\ge -1.

    \begin{gathered}<br />
  P(A \cap B) = P(A) + P(B) - P(A \cup B) \hfill \\<br />
  P(A \cap B) \geqslant 1 - P(A') + 1 - P(B') - 1 \hfill \\<br />
  P(A \cap B) \geqslant 1 - P(A') - P(B') \hfill \\ <br />
\end{gathered}
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