Derive negative binomial probability formula

A question is this type if and only if it requires proving or showing that the probability formula for the rth success equals a given expression involving binomial coefficients.

1 questions · Challenging +1.2

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CAIE FP2 2009 November Q10
10 marks Challenging +1.2
An archer shoots at a target. It may be assumed that each shot is independent of all other shots and that, on average, she hits the bull's-eye with 3 shots in 20. Find the probability that she requires at least 6 shots to hit the bull's-eye. [3] When she hits the bull's-eye for the third time her total number of shots is \(Y\). Show that $$\mathrm{P}(Y = r) = \frac{1}{2}(r - 1)(r - 2)\left(\frac{3}{20}\right)^3\left(\frac{17}{20}\right)^{r-3}.$$ [3] Simplify \(\frac{\mathrm{P}(Y = r + 1)}{\mathrm{P}(Y = r)}\), and hence find the set of values of \(r\) for which \(\mathrm{P}(Y = r + 1) < \mathrm{P}(Y = r)\). Deduce the most probable value of \(Y\). [4]