$x,y,z\in \mathbb{R}$ are chosen at random from the unit interval $[0, 1]$. What is the probability that $$\max(x,y,z) - \min(x,y,z) \leq \frac{2}{3}$$
EDIT- Solutions not using calculus would be appreciated, as this problem appeared on a test where calculus was not meant to be used.
Let $E$ be the event $\{ \max(x, y, z) - \min(x, y, z) \le \frac{2}{3}\}$. Our goal is to find $\Pr[E]$. Now let's split $E$ into 6 events $E_{xy}, E_{yx}, E_{xz}, E_{zx}, E_{yz}, E_{zy}$, where $$E_{xy} = \{ (x = \max(x, y, z)) \land (y = \min(x, y, z)) \land (x - y \le 2/3)\},$$ and the rest are defined similarly. Note that $E$ is the union of these six events and the probability of intersection of any two of these events is $0$. Also, due to symmetry, all these 6 events have the same probability. Hence $\Pr[E] = 6\Pr[E_{xy}]$.
It remains to compute $\Pr[E_{xy}]$. Note that $E_{xy}$ happens iff $y\le x\le y + 2/3$ and $z\in[y, x]$. I.e. \begin{align*} \Pr[E_{xy}] &= \int\limits_0^1 dy \int\limits_y^{\min(y + \frac{2}{3}, 1)} dx \int\limits_y^x dz = \int\limits_0^1 dy \int\limits_y^{\min(y + \frac{2}{3}, 1)} (x - y) dx \\ &= \int\limits_0^1 dy \cdot \frac{(x - y)^2}{2} \Bigg|_{y}^{\min(y + \frac{2}{3}, 1)} = \int\limits_0^1 \frac{\left(\min(y + \frac{2}{3}, 1) - y\right)^2}{2} dy\\ &= \int\limits_0^{\frac{1}{3}} \frac{\left(\frac{2}{3}\right)^2}{2}dy + \int\limits_{\frac{1}{3}}^1 \frac{(1 - y)^2}{2}dy \\ &= \frac{2}{27} -\frac{(1 - y)^3}{6} \Bigg |_{\frac{1}{3}}^1 = \frac{10}{81}. \end{align*}
And thus the answer is $ 6\cdot \frac{10}{81} = \frac{20}{27}$
