Let \(\rho \) be a restriction on \(n\) variables — a map assigning each index \(i \in \{ 0,\dots ,n-1\} \) either a fixed Boolean value or the free mark — and let \(T \subseteq \{ 0,\dots ,n-1\} \). Then the indicator of the event that every index in \(T\) is free under \(\rho \) factors as a product over all \(n\) coordinates:

where the factor \(c_i\) equals \(\mathbf{1}[\rho (i)\ \text{is free}]\) when \(i \in T\) and equals \(1\) otherwise.

Let \(\rho \) be a restriction on \(n\) variables, and let \(U \subseteq \{ 1,\dots ,n\} \) be a set of indices. Then the number of indices in \(U\) at which \(\rho \) is free (unset), viewed as a real number, equals

where \(J\) is the set of free variables of \(\rho \) and each summand is \(1\) when \(\rho (i)\) is unset and \(0\) otherwise.

Fix a real parameter \(p\) and a coordinate \(i\) among the \(n\) variables. Summing the Bernoulli weight of \(\rho \) over all restrictions \(\rho \) on \(n\) variables that leave \(i\) free — that is, those for which \(i\) is a free variable of \(\rho \) — gives

where \(F(\rho )\) denotes the set of free variables of \(\rho \).

Fix \(n\) and two distinct variables \(i \neq j\) among the \(n\) variables, and let \(p \in \mathbb {R}\). Write \(w_p(\rho )\) for the Bernoulli weight of a restriction \(\rho \) on \(n\) variables, namely \(p^{k}\left(\tfrac {1-p}{2}\right)^{n-k}\) where \(k\) is the number of free variables of \(\rho \). Then, summing over all such restrictions, the total weight carried by those \(\rho \) that leave both \(i\) and \(j\) free equals \(p^2\); that is,

Fix \(n\) and a real number \(p\), and let \(\rho \) range over all restrictions on the \(n\) variables, each weighted by its Bernoulli weight \(w_p(\rho )\). Writing \(J(\rho )\) for the set of free variables of \(\rho \), then for every set \(U\) of variables,

Equivalently, when \(0 \le p \le 1\) so that \(w_p\) is a probability distribution, the expected number of free variables landing in \(U\) is \(p\, \left\lvert U\right\rvert \).

Fix \(n \in \mathbb {N}\), a parameter \(p \in \mathbb {R}\), and a set of coordinates \(U \subseteq \mathrm{Fin}\, n\). For a restriction \(\rho \) on \(n\) variables write \(J(\rho )\) for its set of free variables. Summing over all such restrictions, each weighted by its Bernoulli weight, one has

Let \(0 \le p \le 1\), and let \(U \subseteq \mathrm{Fin}\, n\) be a set of coordinates. For a restriction \(\rho \) on \(n\) variables, write \(w_p(\rho )\) for its Bernoulli weight and \(F(\rho )\) for its set of free variables. Then the \(w_p\)-weighted average, over all restrictions \(\rho \) on \(n\) variables, of the squared deviation of \(\left\lvert U \cap F(\rho )\right\rvert \) from \(p\left\lvert U\right\rvert \) satisfies

Fix \(n\) and a parameter \(p\) with \(0 \le p \le 1\), and consider the Bernoulli(\(p\)) random restriction on \(n\) variables, under which each restriction \(\rho \) receives weight \(p^{k}\, \bigl(\tfrac {1-p}{2}\bigr)^{\, n-k}\), where \(k\) is the number of free variables of \(\rho \). For any decidable predicate \(E\) on restrictions, writing \(\Pr [E]\) for the total weight of the restrictions satisfying \(E\), the probability that \(E\) fails equals one minus the probability that \(E\) holds: