Two dice are thrown. $A$: the first die shows an even number. $B$: the sum is 7. Are $A$ and $B$ independent?

Explanation

1. 01

   $P(A) = \frac{1}{2}$, $P(B) = \frac{6}{36} = \frac{1}{6}$, and $A \cap B = \{(2, 5), (4, 3), (6, 1)\}$, so $P(A \cap B) = \frac{3}{36} = \frac{1}{12}$.

2. 02

   $\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$: independent.