Probability that a given number falls between the minimum and the maximum of a sample Let $X$ be a real random variable with absolutely continuous cumulative distribution function $F$. Let $x_{(1)}, ..., x_{(n)}$ be a i.i.d. ordered sample of size $n$ of $X$:
$$
x_{(1)} \leq x_{(2)} \leq ... \leq x_{(n)}.
$$
I am interested by the interval $[x_{(1)}, x_{(n)}]$ defined by the sample minimum and maximum. Let $x \in \mathbb{R}$ be a given arbitrary, constant, number. I want to compute the probability that $x$ falls in the random interval between the minimum and the maximum:
$$
\mathbb{P}\left(X_{(1)} \leq x \leq X_{(n)} \right).
$$
From this post, I can see that:
$$
\mathbb{P}(X_{(1)} \leq x, X_{(n)} \leq y)
= [F(y)]^n - [F(y) - F(x)]^n
$$
for any $x < y$. But this does not finish the proof.

The interval $[x_{(1)},x_{(n)}]$ will not contain $x$ if either all $x_{(i)}$ are below $x$ or all of them are above $x$. Since those are mutually exclusive events the probability of the complement is just
$$ P(X_{(1)} \le x \le X_{(n)}) = 1 - (F(x))^n - (1-F(x))^n$$
