I pick a uniformly random point on a stick of length 1 and sunder the stick there. What is the cumulative distribution function for the ratio of the shorter and longer lengths?
If I let
$S \rightarrow $ shorter length
$L \rightarrow $ longer length
$R \rightarrow $ ratio of shorter and longer lengths ($\frac{S}{L}$)
then
$P(R \le k) = P(\frac{S}{L} \le k) = P(S \le Lk) = P(S \le k(1 - S)) = P(S \le k - kS) = P(S \le \frac{k}{k + 1})$
However, I am not sure where to go from here... any help would be appreciated!