Under the failures-count convention, what distribution describes the number of failures before the first success in independent trials with a fixed success probability?

A suitable rare-event limit links these discrete stories to Poisson counts and exponential waiting times.

Conditions: The context is a rare-event limit.; The variables represent counts and waiting times.

By grouping raw measurements into defined intervals (bins) and tallying the number of observations falling into each bin, a histogram is constructed. The height or area of each bar represents the frequency or count for that interval, visually revealing the regions where data points cluster most densely.

Conditions: The data consists of numerical measurements.; Measurements are sorted into predefined intervals (bins).; The count of observations in each bin is recorded.

Sums of independent exponential waits with the same rate give gamma waiting times.

Conditions: The exponential waits are independent.; All exponential waits share the same rate.

Counting successes in a fixed block of trials gives a binomial variable, while counting failures before a fixed number of successes gives a negative binomial variable.

Conditions: Trials are independent with a fixed success probability.; The first scenario involves a fixed total number of trials.; The second scenario involves waiting for a fixed number of successes.