Suppose there'a tetrahedral dice with numbers $1,2,3,4$. Let's denote by $X$ a rolled number. Prove that $X$ and $5-x$ have the same distribution.
So, for $X$: $$F_X(t) = \mathbb{P}(X \leq t) = \begin{cases} 0, &t < 1\\ 1/4, & 1 \leq t < 2\\ 2/4, & 2 \leq t < 3 \\ 3/4, & 3 \leq t < 3 \\ 1, & 4 \leq t \end{cases} $$ Now, for $Y=5-X$ $$ F_Y(t) = \mathbb{P}(Y \leq t) = \mathbb{P}(5-X \leq t) = \mathbb{P}(X \geq 5 - t) = 1 - \mathbb{P}(X < 5 - t) = 1 - F_X(5-t)^- \\= 1- \begin{cases} 0, &5-t <1\\ 1/4, & 1 \leq 5-t < 2 \\ 2/4, & 2 \leq 5-t < 3\\ 3/4, & 3 \leq 5-t < 4\\ 1, & 4\leq t \end{cases} = \begin{cases} 1, &t > 4\\ 3/4, & 3 < t \leq 4 \\ 2/4, & 2 < t \leq 3\\ 1/4, & 1 < t \leq 2\\ 0, & 1\leq t \end{cases} $$ The problem is that intervals are incorrectly closed. I guess that it is due to the fact that I have $F_X(5-t)^-$, so I cannot explicitly use $F_X(\cdot)$. What shall I do then?