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If $A$ is independent of $B$ and $A$ is independent of $C$ then is $A$ independent of $BC$?

Is this true? Dealing with probability

2 Answers 2

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No.

Think of when $A=B \oplus C$ where $\oplus$ is $XOR$ (Sum modulo 2)

When $B,C$ are independent Bernolli random variables. i.e. are $0,1$ with equal probabilities.

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Here is a simple example using the common coinflipping scenario.

Let our experiment be flipping a fair coin twice in succession. We have an equiprobable sample space: $\{HH,HT,TH,TT\}$

Let $A$ be the event "The first coin is a heads", i.e. the event $\{HH,HT\}$

Let $B$ be the event "The second coin is a heads", i.e. the event $\{HH,TH\}$

Let $C$ be the event "The total number of heads is odd", i.e. the event $\{TH,HT\}$

Each of these occur with probability $\frac{1}{2}$ individually. One can see that these are each pairwise independent as well, $A\cap B, A\cap C,$ and $B\cap C$ each occur with probability $\frac{1}{4}$

However, $A\cap B\cap C=\emptyset$ so $0=Pr(A\cap (B\cap C))\neq Pr(A)\cdot Pr(B\cap C)=\frac{1}{8}$ so we see that $A$ is not independent of $B\cap C$

This example also shows that pairwise independence does not imply mutual independence.