For a probability assignment $P$ on a finite set $X$ of variables, a source query $q$, a $\mathcal{B}[X]$-instance $\hat I$ and a tuple $t$, the marginal probability that $t$ appears in the answer of $q$ on a random world of $\hat I$ equals the probability of the annotation of $t$ in the annotated answer $\langle\!\langle q \rangle\!\rangle_{\hat I}$: $\Pr(t \in q(\hat I)) = \Pr\big(\bigvee_{(t, \alpha) \in \langle\!\langle q \rangle\!\rangle_{\hat I}} \alpha\big)$. This is the paper's Theorem 12, the justification of intensional probabilistic query evaluation: evaluate the query once over Boolean-function annotations and take the probability of the resulting function.

For a probability assignment $P$ on a finite set $X$ of variables, a source query $q$, a $\mathcal{B}[X]$-instance $\hat I$ and a tuple $t$, the marginal probability that $t$ appears in the answer of $q$ on a random world of $\hat I$ equals the probability of the annotation of $t$ in the annotated answer $\langle\!\langle q \rangle\!\rangle_{\hat I}$: $\Pr(t \in q(\hat I)) = \Pr\big(\bigvee_{(t, \alpha) \in \langle\!\langle q \rangle\!\rangle_{\hat I}} \alpha\big)$. This is the paper's Theorem 12, the justification of intensional probabilistic query evaluation: evaluate the query once over Boolean-function annotations and take the probability of the resulting function.

For a probability assignment $P$ on a finite set $X$ of variables, a $\mathcal{B}[X]$-instance $\hat I$, a source query $q$ and a tuple $t$, if $\hat q$ is the query rewritten from $q$ by the rules (R1) to (R4), then $\Pr(t \in q(\hat I)) = \Pr\big(\bigvee_{(t, \alpha) \in [\![\hat q]\!]_{\hat I}} \alpha\big)$: the marginal probability of $t$ is the probability of its annotation in the plain evaluation of $\hat q$ on the composite reading of $\hat I$, each answer tuple read back as an annotated one. This is the paper's Corollary 13, combining the correctness of the rewriting with the theorem on probabilistic evaluation; as for the former, the paper states it with the aggregation rule (R5) included, which this submission does not cover.

The paper's Example 14: each tuple $t_i$ of the $\mathcal{B}[X]$-instance $\hat I$ is kept with an independent probability, $\Pr(t_1) = 0.5$ and $\Pr(t_2) = 0.7$ (the others, unspecified in the paper, are $0.5$ here), and the probability that Nairobi is in the answer of $q_{\mathrm{city}}$ is $\Pr(t_1 \wedge t_2) = 0.5 \cdot 0.7 = 0.35$. Variables are numbered from 0 here, from 1 in the paper.