Probabilities manipulation $P(Z_1+Z_2=2,Z_1=1,Z_2=1)=P(Z_1=1,Z_2=1)$ with $Z_1,Z_2 \in \lbrace 1,2,3,4 \rbrace$ independent and uniform I've two independent and uniform random variables  $Z_1,Z_2 \in \lbrace 1,2,3,4 \rbrace$. Can you tell me why the following equality is correct please?
$$
P(Z_1+Z_2=2,Z_1=1,Z_2=1)=P(Z_1=1,Z_2=1)
$$

This holds even if the variables are neither independent nor uniform because the event $Z_1+Z_2=2$ does not add anything when the RVs are both $1$.
