Grouped Bernoulli failure counts ($\hat{p}={\sum }_j{X}_j/(mq)$)

With $m$ groups of $q$ trials and common marginal failure probability $p$, each count $X_j$ has expectation $qp$, so the displayed estimate is unbiased without assuming independence within groups. Independent individual trials give variance $p(1-p)/(mq)$. If only groups are independent, estimate the variance instead by $s_X^2/(m{q}^2)$, retaining within-group dependence. Equal marginal probabilities alone do not justify a binomial confidence interval.