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

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 $P(Z < 1.5)$?

• 16 minutes. 
– Did
Commented Aug 4, 2013 at 23:10
• @Did: Um, what? Commented Aug 5, 2013 at 1:38

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 $P(Z > 1.5) = 1 - P(Z < 1.5)$.
• In fact, the results are correct for any continuous random variable whose density function is symmetric about $0$, and the assumption of normality is not required in this instance. Commented Aug 5, 2013 at 3:12