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Worked solution

Prove independence after complementing B

The source states that if A and B are independent, then A and B′ are independent. Try proving this result on your own.

CJC · Lecture · Exploration Activity

Partition A using B and its complement

The disjoint events A∩BA\cap B and A∩B′A\cap B' partition A. Hence P(A∩B′)=P(A)−P(A∩B)P(A\cap B')=P(A)-P(A\cap B).

How did this part go?

Use independence and factor

Substitute P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B): P(A∩B′)=P(A)−P(A)P(B)=P(A)[1−P(B)]=P(A)P(B′)P(A\cap B')=P(A)-P(A)P(B)=P(A)[1-P(B)]=P(A)P(B'). This is the product criterion for independence of A and B′. The derivation supplements the source’s “Try proving(2)” prompt.

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Common mistakes

Complement B within the same sample space. This proof uses the product criterion and does not divide by a possibly zero conditional denominator.

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