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## Homework Statement

Let S be the sample space for rolling a single die. Let A={1,2,3,4}, B={2,3,4}, and C={3,4,5}. Which of the pairs (A,B), (A,C), and (B,C) is independent?

## Homework Equations

P(A|B)=P(A)

P(A|B)=P(A&B)/P(B)

P(A&B)=P(A)*P(B)

## The Attempt at a Solution

P(A)=2/3 P(B)=1/2 P(C)=1/2

P(A&B)=P(A)*P(B)=(2/3)(1/2)=1/3 P(A&B)/P(B)=(1/3)/(1/2)=1/6

P(A&C)=1/3 P(A&C)/P(C)=(1/3)/(1/2)=1/6

P(B&C)=1/4 P(B&C)/P(C)=(1/4)/(1/2)=1/8

So based on my calculations, there is none of the pairs which match the independence rule. But the book says that (A,C) is independent.