Calculating upper and lower bounds of the probability of outcome i occuring? Let's say we observe a sample choosing between three possible outcomes: a, b, and c. We can then easily calculate the probability of an individual choosing outcome a: # people choosing a / total # number of people making a choice. 
How can we then add an estimate of the upper and lower bounds of the probability of a occurring?
EDIT: I suppose I could use a bootstrapping procedure: sample from the population (with replacement), make this probability calculation, repeat ~5,000 times, and then measure the 2.5% and 97.5% percentiles to derive a confidence interval on the probabilities. However, perhaps there is a simpler answer out there. Thanks in advance for any suggestions.
 A: the prop.test function in R seems like it would be appropriate. It will take a proportion (e.g. 20 a values drawn from 30 total draws) and calculate the confidence interval around that proportion with the following code:
prop.test(20, 30, correct = FALSE)

Which gives a 95% confidence interval of 48.7% to 80.7% (around a sample value of 66.6%).
A: It looks like you have a plain multinomial problem.
If you are interested only in a confidence interval on each outcome on its own, you can think of all other categories as "not-$i$", reducing it to plain binomial sampling. 
There's a lot of material on binomial proportion confidence intervals.
As such, packages offer CIs for these proportions as a matter of course. Both large and small samples are regularly done.
If you're doing it by hand, the approach of "add 2 to the successes and failures" combined with the usual large-sample interval (i.e. the Agresti-Coull approach at the link above) works surprisingly well even at fairly small sample sizes.
