Binomial cumulative distribution calculation I have exercise like this:
Given probability of success of 0.8 what's probability that in 1000 trials there was more than 800 successes?
So it's quite simple exercise:
It's binomial distribution with distribution function $ f(X = k) = {n \choose k}p^k(1-p)^{n-k} $
So: $ P(X > 800) = (1 - P(X \le 800)) = 1 - \sum_{k=0}^{800} {1000 \choose k}0.8^{k}0.2^{1000-k} = \ ? $
It it possible to calculate it by hand in moderately short amount of time?

This is really a comment too long to fit in 'comment' format:
Computation "by hand" (or with a simple calculator) is not
really feasible, and I'm sure you were intended to use the
normal approximation. Furthermore, I agree with @AndreNicholas
that the normal approximation is good enough for practical
purposes.
However nowadays, exact calculation is feasible using 
software that is as readily accessible as calculators were
a few years ago (maybe when your textbook was
written). As an example, here are some computations using
R software (freely available from r-project.org). Key values agree exactly with those given by @Henry. Notice that
800 is the mean, median, and mode of the distribution Binom(1000, .8).
 n = 1000;  p = .8                # binomial parameters
 1 - pbinom(800, n, p)
 ## 0.4873861                     # exact binomial value

 mu = n*p;  sg = sqrt(n*p*(1-p))  # normal parameters
 1 - pnorm(800, mu, sg)
 ## 0.5                           # normal approx--no continuity correction
 1 - pnorm(800.5, mu, sg)
 ## 0.4842345                     # normal approx--with continuity correction

