I'm trying to understand something that might seem basic to some but its really screwing with my mind.
Imagine a set of $N$ observations of 2 variables. One variable $L$ is one of $A,B,C$ and one $I$ is one of $1,2,3$. My question is in regards to the probabilities. Is it always true that $P(L=A,I=1) = P(L=A)P(I=1)$ ? I just can't seem to generate a dataset where this isn't the case. Any hints as to how I might do so?
Consider a situation where the value of $L$ determines the value of $I$. For example if $L=A$, then $I=1$; if $L=B$, then $I=2$; if $L=C$, then $I=3$. If $L$ is uniform over $\{A,B,C\}$ Then $P(L=A,I=1) = P(L=A) = 1/3 \ne 1/9 = P(L=A) P(I=1)$.