Finding distribution of $V = \frac{X}{2}|X \space even$ where $X \sim Geo(\frac{1}{2})$

Question

Given $X\sim Geo(\frac{1}{2})$ I want to find the distribution of $V = \frac{X}{2}|X \space even$. It is known that $\mathbb{P}(X ~even)=\frac{1-p}{2-p}$, so I tried:
$\mathbb{P}(V=k)=\mathbb{P}(\frac{X}{2}=k|X~even)=\frac{\mathbb{P}(X=2k, ~X~even)}{\mathbb{P}(X~even)}=\frac{(\frac{1}{2})^{2k}}{\frac{1}{3}}=3\cdot(\frac{1}{2})^{2k}\implies V \sim 3\cdot Geo(\frac{1}{4})?$
Does this mean that $V \sim Geo(\frac{3}{4})?$ And if not where is my mistake? (The answer is indeed $V \sim Geo(\frac{3}{4})$)

Answer 1

Geometric distribution:

$$P(V=k)=(1-p)^{k-1}p$$

For your case, $P(V=k)=3\cdot(\frac{1}2)^{2k}=(1-\frac{3}{4})^{k-1}\cdot\frac{3}{4}\Longrightarrow p=\frac{3}{4}$, so it is $V\sim Geo(\frac{3}{4})$