I answered, strictly, what the OP asked: to use the normal approximation to evaluate a binomial problem, NOT what where the different procedures to do it, or what was the best one. The formula to find APPROXIMATELLY p(X<=a) is
______ z = (a + 0.5 - p*N) / s ______ s^2 = p * (1-p) * N
and p(X<=a) = F(z) , the Normal Standard DF at the point z, X~ Binomial (p, N), p the constant probability of success at each Bernouilli N trials. This method is called currently the CLASSICAL CONTINUITY APPROXIMATION, a standard method that was indicated yet in this thread. Jack Tomsky did not take into account. HE DIDN`T PAY ATTENTION (!).