# Comparing more than two proportions within one sample

My experiment was in short: I had less than thirty individuals. One individual had to choose between three colours (green/red/yellow). (Very much like a multiple choice.)

Now, my question is how do I show that significantly more individuals chose red than green or yellow? Can I go with a binary test (one sample, one event) and test for one combination at the time (% that chose green vs % that chose red; % that chose green vs % that chose yellow; % that chose red vs % that chose yellow)?

The simplest way would be to estimate the multinomial probability of an individual choosing each color and estimate confidence intervals. An easy way to do this in R would be with the MultinomialCI package.
library(MultinomialCI)
CI <- multinomialCI(dat$count, alpha = 0.05) dimnames(CI) <- list(dat$color, c("Lower","Upper"))

If the confidence intervals do not overlap, you have a significant difference at the given $\alpha$. If you have individual-level covariates that may influence the selected group, you could use a more complicated method, like a multinomial logistic regression.