Question 25
The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by $$
V=\left\{\begin{gathered} \sqrt2 i,\;-\sqrt2 i, \\ \tfrac{1}{\sqrt8}(1+i), \\ \tfrac{1}{\sqrt8}(-1+i), \\ \tfrac{1}{\sqrt8}(1-i), \\ \tfrac{1}{\sqrt8}(-1-i) \end{gathered}\right\}.
$$ For each $j,$ $1\le j\le12,$ an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P=\prod_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1?$
V=\left\{\begin{gathered} \sqrt2 i,\;-\sqrt2 i, \\ \tfrac{1}{\sqrt8}(1+i), \\ \tfrac{1}{\sqrt8}(-1+i), \\ \tfrac{1}{\sqrt8}(1-i), \\ \tfrac{1}{\sqrt8}(-1-i) \end{gathered}\right\}.
$$ For each $j,$ $1\le j\le12,$ an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P=\prod_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1?$