Example 25.
Continuing from the previous example, \(|\Omega| = 36\text{.}\)
\begin{align*}
A \amp = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\} \\
B \amp = \{(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \\
A \cap B \amp = \{(6, 4), (6, 5), (6, 6)\}
\end{align*}
So \(\Pr(A) = \frac{6}{36}, \Pr(B) = \frac{6}{36}, \text{and } \Pr(A\cap B) = \frac{3}{36}\text{.}\) Then:
\begin{gather*}
\Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{3/36}{6/36} = \frac{3}{6} = \frac{1}{2}.
\end{gather*}
Notice that \(\Pr(A\mid B)\) is significantly larger than \(\Pr(A)\text{.}\)