How come $P(Z< -1.5)$ is equal to $P(Z > 1.5)$ which are both equal to $1-P(Z < 1.5)$?

Question

I can't wrap my head around the idea they are both equal. I mean shouldn't we have $P(-Z > 1.5)$ which is not equal to $P(Z < 1.5)$?

Answer 1

The normal distribution is symmetric about zero, that is why $P(Z < -1.5) = P(Z > 1.5)$ (the two regions of interest are the tails of the distribution). As the normal distribution is a probability distribution, $P(Z \leq 1.5) + P(Z > 1.5) = 1$ (the two regions cover all of the possibile values for $Z$), so $P(Z > 1.5) = 1 - P(Z \leq 1.5)$. As there is zero probability that $Z$ will be precisely $1.5$, $P(Z \leq 1.5) = P(Z < 1.5)$ so $P(Z > 1.5) = 1 - P(Z < 1.5)$.