  A universal cycle, or u-cycle, for a given set of words is a circular word
that contains each word from the set exactly once as a contiguous subword. The
celebrated de Bruijn sequences are a particular case of such a u-cycle, where a
set in question is the set $A^n$ of all words of length $n$ over a $k$-letter
alphabet $A$. A universal word, or u-word, is a linear, i.e. non-circular,
version of the notion of a u-cycle, and it is defined similarly.
  Removing some words in $A^n$ may, or may not, result in a set of words for
which u-cycle, or u-word, exists. The goal of this paper is to study the
probability of existence of the universal objects in such a situation. We give
lower bounds for the probability in general cases, and also derive explicit
answers for the case of removing up to two words in $A^n$, or the case when
$k=2$ and $n\leq 4$.
