Probability of selecting n elements from a continuous uniform distribution outside of given interval

Question

Given a uniform distribution $D$ over $[0,1]$, we are selecting $n$ elements.

What is the probability that none of these elements are for example between $\frac{1}{3}$ and $\frac{1}{2}$?

I know that for one element the probability is $(\frac{1}{2} - \frac{1}{3}) \times \frac{1}{1-0} = \frac{1}{6}$, and hence that element not being in that range will be $1-\frac{1}{6} = \frac{5}{6}$. But I'm not sure how to calculate the probability when I have $n$ such elements.

Answer 1

For each element, their values are completely independent. Therefore, the probablilty is $\frac56\times\frac56\times\cdots\times\frac56=\left(\frac56\right)^n$.