Sample size needed to guess actual probability with high confidence

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Suppose you have a weighted die with n sides. The actual probability of rolling any of those sides face up is varied across all the faces. How many times do you need to roll the die and record the result to know the actual probability of rolling a given face with a 99% confidence level?

I must be doing something wrong with my calculations because the answers I'm getting don't make any sense. Does anyone have advice on how I can model this problem into a formula?