If $X\sim\text{Bin}(3,2)$, find $\Pr(X=k|X\ge 1)$

Question

I'm having some trouble with this. I wrote

$$\Pr(X=k|X\ge 1)=\frac{\Pr(X=k\cap X\ge1)}{\Pr(X\ge1)},$$

but I'm stuck in calculating the quantity $\Pr(X=k\cap X\ge1)$.

Answer 1

If $k \ge 1$, then $P(X=k \cap X \ge 1) = P(X=k)$.

If $k = 0$, then $P(X=k \cap X \ge 1) = 0$.

Answer 2

What about rewriting it as

$$Pr(X = k | X \geq 1) = \frac{Pr(X \geq 1 | X = k) P(X = k)}{Pr(X \geq 1)}$$

That should make it clear, if you think about what is $Pr(X \geq 1|X = k)$.

You should consider $k = 0$ and $k > 0$ separately maybe.