Given an integer $x_0$. Randomly select integers $x_{i+1}\in\{1,\dots ,x_i\}$ uniformly distributed to get a sequence $$x_0\ge x_1\ge\cdots\ge x_{n}=1$$ What is the probability for all of the $x_i$:s, $0<i<n$, to be composites?
Erroneous: There is a claim that this probability is $\frac{1}{x_0}$ but my computationally results suggest a far greater probability.