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In a sphere of radius $R$, $n$ points are distributed randomly and independently from one another. Find the probability that the distance between the center of a sphere and the nearest point from the center is not less than $r,(r<R)$.

What probability methods are useful for this problem (it is obvious that we need to consider the volume of a sphere).

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Let $V(r)$ be the volume of the ball of radius $r$. The probability that a uniformly random point is within $r$ of the center is $$P(≤r)=\frac {V(r)}{V(R)}=\frac {r^3}{R^3}$$ Thus the probability that it is not within $r$ of the center is $$1-\frac {r^3}{R^3}$$

You are asking for the probability that all $n$ of your points are outisde the smaller ball, hence

$$\left(1-\frac {V(r)}{V(R)}\right)^n=\left(1-\frac {r^3}{R^3}\right)^n$$

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  • $\begingroup$ For those wondering, the $\frac{4}{3}\pi$ terms in the volume of a sphere cancel $\endgroup$
    – Henry
    Jan 23 '17 at 13:15

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