Suppose a sphere with radius $R$. Find the probability of the event such that $n$ selected points of the sphere are within the distance of $r = \frac{R}{2}$ from the center of the sphere. The points are uniformly distributed and selected independently.
I tried to proceed using the geometric probability, that is $\mathbb{P}(n \ points \ within \ r) = \mathbb{P}(dist(first) \leq r)\cdot \mathbb{P}(dist(second) \leq r)\cdot...\cdot \mathbb{P}(dist(n) \leq r) = \Bigg(\cfrac{4/3\cdot \pi \cdot r^{3}}{4/3\cdot \pi \cdot R^{3}} \Bigg)^{n} = \bigg(\cfrac{1}{8}\bigg)^{n} \\ \\$
but I have some doubts, as it is possible to choose the same point several times. Is it correct to neglect the events of choosing the same point several times, as we considering the geometric probability?
Since the points are independent and the final condition doesn't correlate them in any way, each point can be treated independently so your answer of $\frac{1}{8^n}$ is correct.