\forall \boldsymbol{S} \subseteq \boldsymbol{V} \backslash\{X, Y, Z\}: A \not \Perp_{P} B \mid \boldsymbol{S} \cup\{X, Y, Z\} \backslash\{A, B\},\forall \bm{S} \subseteq \ensuremath{\bm{V}}\xspace \backslash \{ X,Y,Z \}: A \mathop{\cancel\mathop{\perp\!\!\!\perp}\nolimits}\nolimits_P B \mid \bm{S} \cup \{ X,Y, Z \} \backslash \{ A,B \} \, ,conf 0.683
Y:=((X \oplus Z) \wedge W) \oplus E,
Y := ((X \oplus Z) \land W) \oplus E \; ,
conf 0.877
Y:=((X \oplus Z) \wedge W) \oplus E .
Y := ((X \oplus Z) \land W ) \oplus E \; .
conf 0.877
\begin{aligned} P(X=0, Z=1, Y=1) & =P(X=0, Z=1) \cdot P(Y=1) \\ \Leftrightarrow p+q-2 p q & =q+\frac{p}{2}-p q \\ \Leftrightarrow p-p q & =\frac{p}{2} . \end{aligned}\begin{aligned} P(X = 0, Z=1, Y=1) &= P(X=0,Z=1) \cdot P(Y=1) \\ \Leftrightarrow p + q - 2pq &= q + \frac{p}{2} - pq \\ \Leftrightarrow p - pq &= \frac{p}{2} \; . \end{aligned}conf 1.000
P(X, W \mid Y, Z)=\frac{P(X, W, Y, Z)}{P(Y, Z)} .P(X,W \mid Y,Z) = \frac{P(X, W, Y,Z)}{P(Y,Z)} \; .conf 1.000
\begin{aligned} P(X=1, & W=1, Y=1, Z=1)=\frac{p q}{4}, \\ & P(X=1, Y=1, Z=1)=\frac{q}{4}, \\ & P(W=1, Y=1, Z=1)=\frac{p}{4} . \end{aligned}\begin{aligned} P(X=1, W=1, Y=1,Z=1) &= \frac{pq}{4} \; , \\ P(X=1, Y=1,Z=1) &= \frac{q}{4} \; , \\ P(W=1, Y=1,Z=1) &= \frac{p}{4} \; . \end{aligned}conf 1.000
p q=\frac{p q}{2 P(Y=1)} .pq = \frac{pq}{2 P(Y=1)} \; .conf 1.000
X_{\pi(j)} \not \Perp_{P} X_{\pi(k)} \mid\left\{X_{\pi(1)}, X_{\pi(2)}, \ldots, X_{\pi(k-1)}\right\} \backslash\left\{X_{\pi(j)}\right\},X_{\pi(j)} \mathop{\cancel\mathop{\perp\!\!\!\perp}\nolimits}\nolimits_P X_{\pi(k)} \mid \{ X_{\pi(1)}, X_{\pi(2)}, \dots, X_{\pi(k-1)} \} \backslash \{ X_{\pi(j)} \} \; ,conf 0.797
\begin{aligned} & \pi(1) \rightarrow \pi(4): X \not \Perp_{P} Y \mid\{Z, W\} \\ & \pi(2) \rightarrow \pi(4): Z \not \Perp_{P} Y \mid\{X, W\} \\ & \pi(3) \rightarrow \pi(4): W \Perp_{P} Y \mid\{X, Z\} \end{aligned}\begin{aligned} \pi(1) &\to \pi(4): X \mathop{\cancel\mathop{\perp\!\!\!\perp}\nolimits}\nolimits_P Y \mid \{ Z,W \} \\ \pi(2) &\to \pi(4): Z \mathop{\cancel\mathop{\perp\!\!\!\perp}\nolimits}\nolimits_P Y \mid \{ X,W \} \\ \pi(3) &\to \pi(4): W \mathop{\cancel\mathop{\perp\!\!\!\perp}\nolimits}\nolimits_P Y \mid \{ X,Z \} \ \end{aligned}conf 0.551
\begin{aligned} & \pi(1) \rightarrow \pi(2): X \Perp_{P} Z \mid \emptyset \\ & \pi(1) \rightarrow \pi(3): X \Perp_{P} W \mid Z \\ & \pi(2) \rightarrow \pi(3): Z \Perp_{P} W \mid X . \end{aligned}\begin{aligned} \pi(1) &\to \pi(2): X \mathop{\perp\!\!\!\perp}\nolimits_P Z \mid \emptyset \\ \pi(1) &\to \pi(3): X \mathop{\perp\!\!\!\perp}\nolimits_P W \mid Z \\ \pi(2) &\to \pi(3): Z \mathop{\perp\!\!\!\perp}\nolimits_P W \mid X \; . \ \end{aligned}conf 0.659
\begin{aligned} & \pi^{\prime}(1) \rightarrow \pi^{\prime}(3): X \not \Perp_{P} Y \mid\{Z\} \\ & \pi^{\prime}(2) \rightarrow \pi^{\prime}(3): Z \not \Perp_{P} Y \mid\{X\} \\ & \pi^{\prime}(3) \rightarrow \pi^{\prime}(4): Y \not \Perp_{P} W \mid\{X, Z\} \\ & \pi^{\prime}(1) \rightarrow \pi^{\prime}(4): X \not \Perp_{P} W \mid\{Z, Y\} \\ & \pi^{\prime}(2) \rightarrow \pi^{\prime}(4): Z \not \Perp_{P} W \mid\{X, Y\} \end{aligned}\begin{aligned} & \pi^{\prime}(1) \rightarrow \pi^{\prime}(2): Z \Perp_{P} W \mid \emptyset \\ & \pi^{\prime}(1) \rightarrow \pi^{\prime}(3): Z \Perp_{P} Y \mid\{W\} \\ & \pi^{\prime}(1) \rightarrow \pi^{\prime}(4): Z \not \Perp_{P} X \mid\{W, Y\} \\ & \pi^{\prime}(2) \rightarrow \pi^{\prime}(3): W \not \Perp_{P} Y \mid\{Z\} \\ & \pi^{\prime}(2) \rightarrow \pi^{\prime}(4): W \not \Perp_{P} X \mid\{Z, Y\} \\ & \pi^{\prime}(3) \rightarrow \pi^{\prime}(4): Y \not \Perp_{P} X \mid\{Z, W\} \end{aligned}Z \Perp_{G}\{X, Y\} \mid \operatorname{Pa}(Z) .
X \Perp_{G} Z \mid \operatorname{Pa}(X) \cup(\boldsymbol{X} \backslash\{X\}) \cup(\boldsymbol{Z} \backslash\{Z\}),X \mathop{\perp\!\!\!\perp}\nolimits_G Z \mid \ensuremath{\text{Pa}}\xspace(X) \cup (\bm{X} \backslash \{ X \}) \cup (\bm{Z} \backslash \{ Z \}) \; ,conf 0.713
X \Perp_{G} Z \mid \operatorname{Pa}(X) \cup(\boldsymbol{X} \backslash\{X\}) \cup(\boldsymbol{Z} \backslash\{Z\}),X \mathop{\perp\!\!\!\perp}\nolimits_G Z \mid \ensuremath{\text{Pa}}\xspace(X) \cup (\bm{X} \backslash \{ X \}) \cup (\bm{Z} \backslash \{ Z \}) \; ,conf 0.713
X \Perp_{P} Z \mid \boldsymbol{S} \cup\{Y\} \cup(\boldsymbol{X} \backslash\{X\}) \cup(\boldsymbol{Z} \backslash\{Z\}) .X \mathop{\perp\!\!\!\perp}\nolimits_P Z \mid \bm{S} \cup \{ Y \} \cup (\bm{X} \backslash \{X \}) \cup (\bm{Z} \backslash \{Z \}) \; .conf 0.795