How many samples to be reasonably sure?

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I am new to probability, and I am struggling with the right language to ask or google this question.

I have a population of size $n$ and everyone is say some color. I want to verify that at least half the population is red to within some $\epsilon$ accuracy. How many members of the population do I need to query?

I am not sure which of the main theorems of probability to use - chernoff bound, Markov, central limit theorem, etc.