Find the chance the rth success occurs on trial k. The page traces how the inputs become final-success probability before rounding.
It describes waiting for a sales target, repeated attempts until a quota, or reaching a required event count.
To place the rth success on trial k, the first k−1 trials contain r−1 successes and the final trial succeeds. Negative Binomial Distribution can also be compared with first-success case.
The meaning of final-success probability follows from the relationship, so a different setup can produce a valid answer to a different question.
For r=3,k=8,p=.4, C(7,2).4³.6⁵≈.1045. This Negative Binomial Distribution example can be compared with fixed experiment.
The target and final trial are positive integers with k≥r. Trials remain independent with constant p.
Changing only one field helps distinguish a data-entry problem from the intended behavior of negative binomial distribution.
Choose positions for r−1 early successes, include all success and failure factors, and reserve trial k for success.
Transfer the source entries one at a time under required successes r, trial of rth success k and success probability p, keeping any units visible in your notes. This page can validate numerical ranges, but only the reader can confirm that each entry represents the intended quantity.
At r=1 the formula becomes geometric; increasing k extends the waiting pattern. Trying a nearby input should follow this pattern; otherwise inspect the field assignment.
Binomial fixes trial count and counts successes; negative binomial fixes a target and asks when it arrives. That neighboring measure needs its own setup rather than a relabeled answer.
Attach units and the name final-success probability whenever the answer leaves this page. A rounded value is suitable for presentation, while retained working digits are safer for a dependent calculation.
Copying only the decimal discards the setup. Pair the final-success probability with required successes r, trial of rth success k and success probability p and the unit convention so another reader can reconstruct its meaning.
For a small Negative Binomial Distribution example using Required successes r, Trial of rth success k, Success probability p, solve once with the displayed formula and once through direct enumeration or a probability tree. Two methods that organize outcomes differently are less likely to share the same hidden assumption error.
For Negative Binomial Distribution, when the exact list becomes too large, retain a simpler identity as a check: complementary probabilities sum to one, a Pascal row is symmetric, and adjacent factorial results have a known ratio. State which identity was used beside the reported answer.
The result is geometric.
The kth trial is fixed as success.
No.
The per-trial probability.