Proving complement of unions equals intersection of complements.

[cookiesyum]

Homework Statement

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

' = complement

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

Homework Equations

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

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

The Attempt at a Solution

 

[statdad]

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

$$(C_1 \cup C_2)' = C_1' \cap C_2'$$

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

$$\Pr(C_1 \cup C_2 \cup \cdots \cup C_k) = 1 - \Pr((C_1 \cup C_2 \cup \cdots C_k)') = 1 - \Pr(C_1' \cap C_2' \cap \cdots \cap C_k')$$

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