Probability of satisfaction of a collection of words

For a finite group and a single word
Suppose $$G$$ is a finite group and $$w$$ is a word in $$n$$ letters. $$w$$ defines a word map from $$G^n$$ to $$G$$ by sending any $$n$$-tuple in $$G$$ to the word $$w$$ evaluated at that $$n$$-tuple. The probability of satisfaction of $$w$$ is the probability (under the uniform distribution on $$G^n$$) that the image of the word map gives the identity element of $$G$$.

Explicitly, denote by $$S_w$$ the subset of $$G^n$$ given by:

$$S_w = \{ (g_1,g_2,\dots,g_n) \mid w(g_1,g_2,\dots,g_n) = e \mbox{ (the identity element of } G \mbox{)} \}$$

Then, the probability of satisfaction of $$w$$ is the quotient:

$$\frac{|S_w|}{|G|^n}$$

For a finite group and a collection of words
Suppose $$G$$ is a finite group and $$C$$ is a collection of words, all of which use a fixed alphabet of $$n$$ letters. Thus, each $$w \in C$$ defines a word map from $$G^n$$ to $$G$$ by sending any $$n$$-tuple in $$G$$ to the word $$w$$ evaluated at that $$n$$-tuple. The probability of satisfaction of $$C$$ is the probability (under the uniform distribution on $$G^n$$) that the image of the word map for every $$w \in C$$ gives the identity element of $$G$$.

Explicitly, denote by $$S_C$$ the subset of $$G^n$$ given by:

$$S_C = \{ (g_1,g_2,\dots,g_n) \mid w(g_1,g_2,\dots,g_n) = e \mbox{ (the identity element of } G \mbox{)}  \ \forall \ w \in C\}$$

Then, the probability of satisfaction of $$C$$ is the quotient:

$$\frac{|S_C|}{|G|^n}$$

Facts for probability of satisfaction applicable to single words and to collections of words

 * The probability of satisfaction of $$w$$ in $$G$$ is at least $$1/|G|^n$$ where $$n$$ is the number of letters appearing in $$w$$. This is because the tuple with all coordinates the identity element must satisfy the word.
 * Probability of satisfaction is invariant under Andrews-Curtis transformations: The probability of satisfaction of a word is invariant under conjugation. It is also invariant under performing Nielsen transformations on the word in the free group on $$n$$ letters. Thus, it is invariant under Andrews-Curtis transformations. The same result holds for a collection of words, but we need to do the Andrews-Curtis transformations on the subset together rather than doing a different transformation on each word.
 * The probability of satisfaction of $$w$$ in $$G$$ is 1 if and only if $$G$$ is in the subvariety of the variety of groups where $$w$$ is satisfied (i.e., where $$w$$ equals the identity element). Similarly, the probability of satisfaction of a collection $$C$$ of words in $$G$$ is 1 if and only if $$G$$ is in the subvariety of the variety of groups where all $$w \in C$$ are satisfied.
 * Words defining the same variety may have different probability of satisfaction: Note that even if two words define the same subvariety of the variety of groups, the probability of satisfaction of the two words may be different. For instance, the word $$x^2$$ (in one letter) and the word $$x^2y^2$$ (in two letters) define the same subvariety -- the variety of elementary abelian 2-groups. However, in the quaternion group, the former word has probability of satisfaction 1/4 whereas the latter word has probability of satisfaction 5/8.

Facts for probability of satisfaction applicable only to single words
If there exists a letter in the word that appears just once in the word with an exponent of 1 or -1, then the probability of satisfaction of the word is $$1/|G|$$. The reason is that for any choice of values of the other letters, there will always be a unique choice of that letter for which the word is the identity element. Thus, for instance, the words $$xy,xy^{-1},x^2y^{-1}, xy^{-1}x,xyx^{-1},y^2xz^{-3}$$, all have probability of satisfaction $$1/|G|$$.