Two dice are thrown. The events $A, B$ and $C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\leq 5$.
Describe the event $A^{\prime}$.

(B) When two dice are thrown,the sample space $S$ contains $36$ outcomes:
$S = \{(x, y) : x, y \in \{1, 2, 3, 4, 5, 6\} \}$.
The event $A$ is defined as getting an even number on the first die:
$A = \{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \}$.
The complement event $A^{\prime}$ consists of all outcomes in $S$ that are not in $A$. This means the first die must show an odd number:
$A^{\prime} = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \}$.
Comparing this with the definition of event $B$ (getting an odd number on the first die),we see that $A^{\prime} = B$.