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Proving the inclusion-exclusion principle

  1. Sep 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove the general inclusion-exclusion formula.

    2. Relevant equations

    Inclusion-exclusion Formula:

    P(C1 U C2 U...U Ck) = p1 - p2 + p3 - ... + (-1)(k+1)pk

    where pi equals the sum of the probabilities of all possible intersections involving i sets.

    3. The attempt at a solution

    Assume that the formula holds for n sets. Then for n+1 sets...

    P(union k=1 to n+1 of Ck) = P((union k=1 to n of Ck) union Cn+1) = P(union k=1 to n of Ck) + P(Cn+1) - P((union k=1 to n of Ck) intersect Cn+1) = P(union k=1 to n of Ck) + P(Cn+1) - P(union k=1 to n of (Ck intersect Cn+1)) = sum k=1 to n of P(Ck) + P(Cn+1)....

    From there I don't know how to show the alternating pattern.
     
    Last edited: Sep 30, 2009
  2. jcsd
  3. Sep 30, 2009 #2

    Office_Shredder

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    You can use your inductive hypothesis to write out

    P(union k=1 to n of Ck)

    and

    P((union k=1 to n of Ck) intersect Cn+1)

    Using the fact that Cn+1 intersected with the union from 1 to n of Ck is the same as the union from 1 to n of (Ck intersected with Cn+1)
     
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