\begin{align*} P(A) & = P( \{ a_1 \} \cup \{ a_2 \} \cup ... \cup \{ a_{|A|} \} )\\ & = P(\{ a_1 \}) + P(\{ a_2 \} ) + ... + P(\{ a_{|A|} \} )\\ & = \frac{1}{{|S|}} + \frac{1}{{|S|}} + ... + \frac{1}{{|S|}}\\ & = \frac{|A|}{{|S|}} \end{align*}