Probability of a Uniform Random Variable

Question

Let $X_1, X_2, X_3$ be iid Uniform (0,1) random variables.

How do I find the probability that $X_{\min} = \min[X_1,X_2,X_3]$, is between 0 and 1/2?

Answer 1

For the min not to be $\le 1/2)$, all have to be $\gt 1/2$.

$P(X_{min}\le 1/2)=1-P(X_{min}\gt 1/2)=1-\prod_{k=1}^3 P(X_k\gt 1/2)=1-1/8=7/8$

Answer 2

Since $X_{i} \sim U(0, 1)$, we have $P(X_{i} \leq 1/2) = 1/2$ for symmetry reasons. Thus,

$$P(X_{min} \leq 1/2) = 1 - P(X_{1}, X_{2}, X_{3} \geq 1/2) $$

$$= 1 - P(X_{1} \geq 1/2) \cdot P(X_{2} \geq 1/2) \cdot P(X_3 \geq 1/2) = \boxed{\frac{7}{8}}$$