show that if events A and B are independent, then
1.event A' and B are independent
2.event A' and B' are independent
please step by step
A and B are independent ⇒ P(A∩B)=P(A)P(B)
1. P(B)=P[B∩(A∪A')]
⇒P(B)=P[(B∩A)∪(B∩A')]
⇒P(B)=P(B∩A)+P(B∩A')-P(B∩A∩B∩A')
⇒P(B)=P(B∩A)+P(B∩A')
⇒P(B)=P(B)P(A)+P(B∩A')
⇒P(B)-P(A)P(B)=P(B∩A')
⇒P(B∩A')=P(B)[1-P(A)]=P(B)P(A')
∴A' and B are independent
2.P(A'∩B')
=P(A∪B)'
=1-P(A∪B)
=1-P(A)-P(B)+P(A∩B)
=1-P(A)-P(B)+P(A)P(B)
=(1-P(A))(1-P(B))
=P(A')P(B')
∴A' and B' are independent
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Verified answer
A and B are independent ⇒ P(A∩B)=P(A)P(B)
1. P(B)=P[B∩(A∪A')]
⇒P(B)=P[(B∩A)∪(B∩A')]
⇒P(B)=P(B∩A)+P(B∩A')-P(B∩A∩B∩A')
⇒P(B)=P(B∩A)+P(B∩A')
⇒P(B)=P(B)P(A)+P(B∩A')
⇒P(B)-P(A)P(B)=P(B∩A')
⇒P(B∩A')=P(B)[1-P(A)]=P(B)P(A')
∴A' and B are independent
2.P(A'∩B')
=P(A∪B)'
=1-P(A∪B)
=1-P(A)-P(B)+P(A∩B)
=1-P(A)-P(B)+P(A)P(B)
=(1-P(A))(1-P(B))
=P(A')P(B')
∴A' and B' are independent