Let $X_1$ and $X_2$ be two discrete random variables (both $X_1$ and $X_2$ Poisson law and not necessarily independent) defined over a probability space, denoted by $(\Omega, \mathcal{F},\mathbb {P})$.
We know that we can write that :
$$ \mathbb{E}[X_1\;|\, X_2 \geq 1 ] = \frac{1}{\mathbb{P}[X_2 \geq 1]} \sum_{i=0}^\infty \, i\,\mathbb{P}[\{X_1 = i\}\, \cap \, \{X_2 \geq 1\} ]$$
We know that when $X_1$ and $X_2$ are independent, we can write that $$\mathbb{P}[\{X_1 = i\}\, \cap \, \{X_2 \geq 1\}] = \mathbb{P}[\{X_1 = i\}] \times \mathbb{P}[\{X_2 \geq 1\}]$$
Is there a more explicit way of writing the quantity $$\mathbb{P}[\{X_1 = i\}\, \cap \, \{X_2 \geq 1\}]. $$ I tried the following computations: $$\mathbb{P}[\{X_1 = i\}\, \cap \, \{X_2 \geq 1\}]= \mathbb{P}[\{X_1 = i\}\ \cap \bigcup_{k\geq 1}\{X_2 = k\}]= \mathbb{P}[\bigcup_{k\geq 1}\left(\{X_1 = i\}\ \cap \{X_2 = k\}\right)],$$ which is not very informative.