calculate a probability of a Poisson process

Question

Let $\{X_t\}_{t\geqslant 0}$ be a Poisson process with parameter $\Lambda\geqslant 0$. Calculate $\mathbb{P}[X_1\gt 1 │ X_3 = 5]$.

I was thinking to calculate this with definition conditional probability but i don't understand this: $\mathbb{P}[X_1 \gt 1 , X_3=5]$.How can I see this?

Answer 1

HINT

This is a union of all events $\{X_1 = a, X_3 = 5\}$ where $a > 1$. In other words, $$ \{X_1>1, X_3 = 5\} = \bigcup_{a>1} \{X_1 = a, X_3 = 5\} $$