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A universal cycle, or u-cycle, for a given set of words is a circular wordthat contains each word from the set exactly once as a contiguous subword. Thecelebrated de Bruijn sequences are a particular case of such a u-cycle, where aset in question is the set $A^n$ of all words of length $n$ over a $k$-letteralphabet $A$. 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 forwhich u-cycle, or u-word, exists. The goal of this paper is to study theprobability of existence of the universal objects in such a situation. We givelower bounds for the probability in general cases, and also derive explicitanswers for the case of removing up to two words in $A^n$, or the case when$k=2$ and $n\leq 4$.

On the probability of existence of a universal cycle or a universal word for a set of words