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Proving complement of unions equals intersection of complements.

  • Thread starter cookiesyum
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Homework Statement



Generalize to obtain (C1 U C2 U...U Ck)' = C1' intersect C2' intersect...intersect Ck'

' = complement

Say that C1, C2,...,Ck are independent events that have respective probabilities p1, p2, ..., pk. Argue that the probability of at least one of C1, C2,...,Ck is equal to 1 - (1-p1)(1-p2)...(1-pk)

Homework Equations



I don't know how to generalize that...

For the second part, P(C1 U C2 U...U Ck) = 1- P(C1 U C2 U...U Ck)' = 1 - P(C1' intersect C2' intersect...intersect Ck') = 1 - (1-p1)(1-p2)...(1-pk). Not sure how that proves at least one of Ck has to equal that though...

The Attempt at a Solution

 

Answers and Replies

  • #2
statdad
Homework Helper
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for the first part why not begin trying to show it is true for k = 2: that is, try to show

[tex]
(C_1 \cup C_2)' = C_1' \cap C_2'
[/tex]

Once you have it for k=2, use induction for the general case.

for the second part (once the first is shown) your first line should read

[tex]
\Pr(C_1 \cup C_2 \cup \cdots \cup C_k) = 1 - \Pr((C_1 \cup C_2 \cup \cdots C_k)') = 1 - \Pr(C_1' \cap C_2' \cap \cdots \cap C_k')
[/tex]

At this point, use the facts that [tex] \Pr(C_j) = p_j [/tex] (so you know the probabilities of the complements) as well as the fact that the events are independent.
 

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