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Sample size for a variable number of answers

I have data which was generated from a (theoretical) multinomial distribution, and using the data points I estimate the multinomial probabilities. Ideally, I would like to know how many points i would need be able to be confident that the inferred multinomial distribution is correct, up to +-0.01 bias, with probability 99% - i.e. i would like to know how many points are needed to get a small confidence interval of +- 0.01. The categorical distribution behind the multinomial has 4 values, and i can safely assume that p1...p4>0.1

How shall i go about achieving this? Am i looking to do some sort of chi-squared test?

I'm a bio-informatist with relatively meager knowledge in statistics, so I apologize in beforehand if the question is too basic - If it is, I'd be glad to be referred to some relevant ( hopefully not too long ) reference. Thanks!