Book question: If $Z \sim N(0,1)$ and $P(Z<k) = \Phi(k),$ for $k>0$, find $P(|Z|> k)$ in terms of $\Phi(k)$.
I thought this would be straightforwardly just $2(1-\Phi(k))$. But the answer in the back is $2\Phi(k)-1$, which I don't get at all. Surely $2\Phi(k)-1 = P(|Z|< k)$ ?
You are absolutely right! Your textbook answer is referred to the complement probability
$$\mathbb{P}[|Z|<k]$$