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if P(A)>0 then P(B'|A)=1-P(B|A)

so P(A intersect B')=P(A)P(B'|A)=P(A)[1-P(B|A)]

=P(A)[1-P(B)]

=P(A)P(B')

the definition of independent events is if P(A intersect B')=P(A)P(B')

that is the proof

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Q: If A and B are independent events then are A and B' independent?
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If A and B are independent events than A and B' are independent?

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