Does independence from independent events imply joint independence?

40 Views Asked by At

Take three events $A,B,C$. Is it true that if $A\perp B$, $A\perp C$, and $B\perp C$, then $A\perp (B,C)$?

I don't see how to prove this. I stopped here: $$ p_{A|B,C}=\frac{p_{A,B,C}}{p_{B,C}}=\frac{p_{A,B,C}}{p_{B}p_{C}}=???=p_A $$