Explanation

(a)
Given, $E_{k} = \left\{ (a,b) \in S:ab = k \right\} for\ k \geq 1\ and\ p_{k} = P\left( E_{k} \right)$
Now, $E_{1} = \left\{ 1,1 \right\}\ \Rightarrow \ p_{1} = P(E_{1})$
$\Rightarrow \ P_{1} = \frac{1}{36}$
$E_{2} = \left\{ (1,2),(2,1) \right\}\ \Rightarrow \ p_{2} = P(E_{2})$
$\Rightarrow \ P_{2} = \frac{2}{36}$
$E_{4} = \{(1,4),(4,1),(2,2)\}$
$\Rightarrow \ P_{4} = P(E_{4})$
$\Rightarrow \ P_{4} = \frac{3}{36}$
$E_{6} = \{(1,6),(6,1),(2,3),(3,2)\}$
$\Rightarrow \ P_{6} = P\left( E_{6} \right)\ \Rightarrow \ P_{6} = \frac{4}{36}$
and $E_{30} = \left\{ (5,6),(6,5) \right\}\ \Rightarrow \ p_{30} = p(E_{30})$
$\Rightarrow \ p_{30} = \frac{2}{36}$
$\therefore$ From the above results, we get
$p_{1} < p_{30} < p_{4} < p_{6}$