>> the uncertainty in the number of trials > Has no meaning to me.
What the author is trying to get at in the admittedly poorly worded question is that the trials are noisy measures of an underlying effect. Your job is to sort by effect size, while accounting for the random chance that a low sample size trial just got unlucky.
You might argue that the question is much harder than the author assumes, since your best guess at the actual effect size seems like it should still just be the success rate, even if the low sample size trials have wider error bars. You'd need to come up with some sort of heuristic that says why 7/9 deserves a lower rank than 50/70 using binomial confidence intervals.
Probably that heuristic is intended to be a bayesian approach? Like, if you add just two successes and two failures to each scenario as a prior, thats enough to put the 50/70 option ahead.
The essence of my comment was that this text/test is not for me (one person of the general public) but more like a few leetcode-style questions for statisticians.
“However, it is very important that the uncertainty in the number of trials is taken into account because over-estimating a fraction is a costly mistake.“
Seems fairly clear to me that you’re supposed to use a lower bound estimate to take into account variance on the fraction due to the number of trials in a way to bounds the chance of over estimation.
Further, there is no need for a heuristic when there a several statistical models for this exact problem with clear properties. Some are given in the answer.
> The lower bound of which can be used to order the fractions, and so control the risk of over-estimation.
It not clear to me from the question whether the cost of a mistake is in the over-estimating the underlying effect or in misranking the effects, and that seems like it would drive your heuristic selection.