How to use a probability under 0.5000 with the normal distribution table?

Question

When you are given P( Y < y )= probability x where Z is normally distributed and you know the value of x and the mean and variance it is realitively simple to calculate y.

However if x is under 0.5000 it isn't on the normal distribution table/chart.

I always though in this situation you simply did 1-phi(1-x).

However my calculations are coming out wrong.

Is my methord wrong?

Answer 1

If random variable $Y$ has normal distribution with mean $0$, then $$P\{Y \in B\}=P\{Y \in -B\}$$ for every measurable set $B$, since normal distribution is symmetric about it's mean (recall that it's density, the so called bell curve, is symmetric with respect to $y$-axis). As a consequence, $$P\{Y<0\}=P\{Y\ge 0\}=\frac{1}{2}$$ and, for $y<0$ $$\begin{aligned} P\{Y-y\}&=1-P\{Y \le -y\}\\ &=1-P\{Y < -y\}.\end{aligned}$$ The last equation holds since $Y$ is continuous random variable.