For p ≤ 0.5

There are tables in most formulae books or in the back of a statistics text book that allow you to calculate some Binomial probabilities with ease. This is outlined in the following tutorial.

Binomial Distribution - Cumulative Probability Tables : ExamSolutions - youtube Video

For p > 0.5

There are tables in most formulae books or in the back of a statistics text book that allow you to calculate some Binomial probabilities with ease. However, sometimes the tables only give values of p up to 0.5.
This tutorial explains how to use the tables still when p>0.5.

Binomial Distribution : Cumulative Probability Tables (2) : ExamSolutions - youtube Video