Incidentally, the normal approximation to the binomial gives the value 0.183472 which, compared to the 0.195258 obtained directly, has a relative error of 6.4%. Too large, maybe..., depending, of course, on the purpose of the exercise. But, of course, it takes almost the same time and effort to calculate both results. So, come to think of it, nowadays the central limit theorem is perhaps as outdated as the tables of logarithms (for numerical approximations purposes).

I obtain the result in less than 6 seconds (rather long, but not really complicated)

probability 0 <= x <= 199 if x is binomial with n = 629 and p = 1/3

2790202042683170027396338572156271486236764921820554406324053343438420\ 7473524012382076372430640929027629909848334225903726967528396496546094\ 2768804990201714967763531874391873967728927521481203010698948463470386\ 5556684010030811850825222228971811003257372984250289096178370247660188\ 8824446582320529408/\ 1428981838579134870963576158997950040247615937024229320041839443209485\ 7063291801552419719892590341365005232809905728568571532045291896518962\ 0416249357065663843887173052694852955236403957825594931793818273072298\ 1380571684806205514443614319046207595729139902191275491143892128100897\ 71517714979850768987

In[3]:= N[%]

Out[3]= 0.195258

Thanks Gianluca, no, I haven't put a semicolon at the end. I tried to solve the Problem wit my handy and it works. Best regards Joachim