To get the A on the last pick requires that all of the previous picks to be something else. You don’t get the opportunity to pick the A if it has already been selected. So, if L stands for losing (not getting the A), then
\begin{align*}
P(\text{last}) & = P(\text{LLLLLLLLLLLLLLA}) \\
& = \frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2} = \frac{1}{15}.
\end{align*}
Therefore, it is the same probability of getting the A whether you pick first or last. In general, to win on the kth pick gives
\begin{align*}
P(\text{kth}) & = P(\text{LL...LA})\\
& = \frac{14 \cdot 13 \cdot ... \cdot (15-k) \cdot 1}{15 \cdot 14 ... \cdot (16-k) \cdot (15-k)} = \frac{1}{15}
\end{align*}
Hence, it is the same probability regardless of when you get to pick.