Negative Binomial Distribution
The negative binomial distribution describes the number of trials X required to achieve r successes when each trial has probability p of success. The probability is P(X = k) = C(k-1, r-1) × p^r × (1-p)^(k-r) for k = r, r+1, r+2, ... When r=1, it reduces to the geometric distribution.
Real World
A quality control engineer at Jaguar Land Rover tests engine components off the production line and uses the negative binomial distribution to model how many units must be inspected before finding the third defective part, helping plan inspection schedules.
Exam Focus
Show that when r = 1 the negative binomial reduces to the geometric distribution — examiners often test whether you recognise this link.
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