Geometric variable at most 8 given at least 5

Let $X$ be geometric with success probability $\tfrac14$, counting the trial on which the first success occurs (so $X\in\{1,2,\dots\}$). Find $\mathbb{P}[X\le 8\mid X\ge 5]$.

$\tfrac{175}{256}$. Knowing $X\ge5$ means the first four trials failed; by memorylessness we need a success within the next four trials. That fails with probability $(\tfrac34)^4=\tfrac{81}{256}$, so the answer is $1-\tfrac{81}{256}$.