Cont. Uniform Dist. Problem with Independent Random Vars

Question

Let $X_1$, $X_2$, and $X_3$ be independent random variables with the continuous uniform distribution over $\left[0,2\right]$. Let $Z = \min\left(X_1, X_2, X_3\right)$.

Find $\mathbb{P}\left(Z \geq 0.5\right)$.

I had thought this was $\left(\frac{1}{4}\right)^3 = \frac{1}{64}$, but my answer was incorrect. Could anyone help me reach the correct answer with a short explanation?

Answer 1

If $Z=\min X_n$, then $P(Z \geq r)=P(X_1 \geq r)\cdot P(X_2 \geq r) \ldots P(X_n \geq r)$. Do you see why?