In statistics, how do you calculate the probability associated with a certain Z score? Tables don't give P values for Z scores above 3. ?

find a more extensive table. Abramowitz & Stegun. Table 26.1, 

p 972

P(3.00)

= .99865 01020

P(5.00)

= .99999 97133

And table 26.2 large arguments. 

- log Q(x)

5 6.54265

When it is normalised, ie μ = 0, σ = 1, and you work directly with z, then

pdf(z) = 1/√(2π) e^(-½ z²)

Then you integrate that between the particular limits of z to get the probabilities you want (good luck with that). Note that the tails stretch to ±∞ but since the mean is centered on zero we know that each side has an area of 0.5. So for instance, p(4) is 0.5 + the integral from 0 to 4.

Basically, you either use WolframAlpha or you track down some tables where someone has already done the work, and probably gives the results as p(>z) so that it is just a tiny number in scientific form, rather than 0.9 followed by a ton of 9's.

You can lookup in the ztable and find the corresponding area. 

The "exact" way to find the values (or to create the table) is by evaluating the integral of the PDF from z=a to z=b.

Such an integral can't be exactly found in closed form (except for some specific values) and numerical approximations are thus used. 

For more info, see

https://en.wikipedia.org/wiki/Normal_distribution