Let $O$ be a set of outcomes $O_1, O_2, ... O_n$ with [positive] probabilities $P_1, P_2, ... P_n$ in $P$, with $\sum P_i = 1$. How many outcomes, $f(K)$, would you need to be $k$% sure that the probability distribution described by set $P$ is correct?
If it's unclear what I mean, here's an example question:
Suppose I have a gumball machine, and on the owner's manual it says that the machine dispenses red gumballs 15% of the time, yellow gumballs 15% of the time, green gumballs 30% of the time, and blue gumballs 40% of the time.
How many gumballs would I need as (a function of x) such that I could be x% sure of the above distribution probabilities?
Sorry if this is a simple question. The only statistics I have done are simple z-scores, t-tests and the like.
Edit: By "x% confident," I mean that one would expect x% of confidence intervals for the probabilities of the events to contain the true probabilities.