No two samples too close

HardProbability Combinatorics~10m

You sample $101$ IID Uniform $(0,1)$ values. Find the probability that no two of them lie within a distance of $\tfrac{1}{1000}$ of each other. The answer reduces to the form $\left(\tfrac{a}{b}\right)^c$ ; report $a + b + c$ .

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For $n$ uniforms on $[0,1]$ , $\mathbb{P}[\text{min gap}\ge d]=(1-(n-1)d)^n$ .
$n=101,\ d=\tfrac1{1000}$ : $(1-\tfrac1{10})^{101}=(9/10)^{101}$ , so $9+10+101$ .

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Asked at: Jane Street, Two Sigma

No two samples too close

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