Derive negative binomial probability formula

10 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. When she hits the bull's-eye for the third time her total number of shots is \(Y\). Show that $$\mathrm { P } ( Y = r ) = \frac { 1 } { 2 } ( r - 1 ) ( r - 2 ) \left( \frac { 3 } { 20 } \right) ^ { 3 } \left( \frac { 17 } { 20 } \right) ^ { r - 3 } .$$ 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 \(Y\).