Confidence interval probability

Question

I am struggling to even understand how to approach this problem

Finding the probability of $$P(Z<z_a+z_{1-a})$$

Where Z is a standard normal variable and

$$P(Z\leq z_a)=1-a$$

Is there manipulation rules for adding arguments in a normal distribution? How do I even start with this?

Answer 1

There's an implicit symmetry here: note that $z_a = -z_{1-a}$ because $\Pr(z>z_a)=a$ also $\Pr(z<z_{1-a}) = a$. This is telling you that the area to the left of $z_{1-a}$ is equal to that on the right of $z_a$. With that being said,

$$\Pr(Z < z_a + z_{1-a}) = \Pr( Z <0) = 0.5$$