Approximate formulae are derived for obtaining confidence intervals for the probabilities of a multinomial distribution. The approach used is to consider the Chi-square goodness of fit statistic as a function of the population parameters and to invert this function to obtain a set of simultaneous confidence intervals for the parameters

The confidence coefficient for the set of simultaneous confidence intervals obtained by this procedure is conservative, i.e., the true probability that every interval covers its corresponding parameter will in general be greater than the coefficient obtained by this method. As the sample size increases the intervals will converge on the population parameters and will estimate them exactly in the limit.