Imagine I'm performing an experiment where I treat cells with some compound looking for toxicity. I test $N$ identical cells with the compound and all cells die immediately. As a function of $N$, how certain can I be that the next sampled cell will die when treated with the compound?
2026-04-07 04:43:35.1775537015
How do we estimate binomial confidence intervals when there are zero successes or failures?
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If what you’re looking for is the standard binomial confidence interval, then answer is P≥alpha^(1/N). (If the cells are i.i.d. with P the probability that a cell will die, then the probability that N out of N died is Pr[X=N]=P^N. A confidence interval is a list of all parameter values supported by the data. If we decide support is that the chance of observing what we did has to alpha or higher, then the lower confidence limit is all P such that P^N≥alpha, which can be rewritten P≥alpha^(1/N). For example, if N=50, then the lower confidence limit is P≥0.94.)