The probability for a finite sequence to not include a prime 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.
 A: [Sorry this is not an answer, but more like a discussion.  I am new here and unsure how to post something like this, since it wont fit in the comments...]
The way I understand the problem statement, if some $x_i$ happens to be very small then the rest of the chain will be rather short, i.e. $n-i$ will also be small.  So it is not just a matter of whether the next number is non-prime but also its size is critical.  This suggests a recurrence (dynamic programming) to me.
Let $p(x)$ denote the probability of starting at $x_0=x$ and having all $x_i: 0 < i \le n$ be non-prime (so this allows $x_n=1$, which is non-prime).  A recurrence can be written by considering the next number picked, $y$.  First, each $y \in \{1,..., x\}$ is picked with uniform probability $\frac{1}{x}$.  Second, if you pick a non-prime $y$ you can continue, whereas if you pick a prime $y$ you have failed.  So the recurrence would be: 
$$p(x) = \frac{1}{x} \sum_{non\ prime\ y \le x} p(y), \forall x$$
The above works for both prime and non-prime $x$, and in case $x$ is prime, the $\le$ sign can be replaced with $<$.  In case $x$ is not prime, then $p(x)$ appears on both side of the equation and cancelling the term is helpful: 
$$p(x) = \frac{1}{x-1} \sum_{non\ prime\ y < x} p(y), \forall\ non\ prime\ x$$
All summations include $y=1$ and we also need $p(1)=1$.  
I am not sure how to proceed from this recurrence, e.g. to seek a closed-form solution or approximation, but it can be iterated easily: $p(1)=1, \ p(2)=\frac{1}{2},\  p(3)=\frac{1}{3}, \ p(4)=\frac{1}{3}, \ p(5)=\frac{1}{5}(p(1)+p(4)) = \frac{4}{15}$ etc.
Am I on the right track?
