What does “$A_{ij}$ are two by two independent but not mutually independent” mean”

Question

I'm doing this exercise in probability class:

I could not understand what $A_{ij}$ are two by two independent but not mutually independent means. From my understanding, we talk about independence when we have at least two random variables. Clearly, $A_{ij}$ is an event, not a variable in this case.

Thank you for your help!

Answer 1

Two events $A,B$ are independent iff $P(A\cap B)=P(A)P(B)$. So you must show that $P(A_{ij}\cap A_{i'k'})$ breaks up into a product whenever $i\neq i'$ and $j\neq j'$. However $P(A_{12}\cap A_{13}\cap A_{23})$ doesn't factor.