The random variable X has the negative binomial distribution NB(5, p), where p < 0.5, and ${\text{P}}(X = 10) = 0.05$. By first finding the value of p, find the value of ${\text{P}}(X = 11)$.

${\text{P}}(X = 10) = \left( {\begin{array}{*{20}{c}}   9 \\   4 \end{array}} \right){p^5}{(1 - p)^5}$     (= 0.05)     (M1)A1A1

Note: First A1 is for the binomial coefficient. Second A1 is for the rest.

solving by any method, $p = 0.297 \ldots$     A4

Notes: Award A2 for anything which rounds to 0.703.

Do not apply any AP at this stage.

${\text{P}}(X = 10) = \left( {\begin{array}{*{20}{c}}   {10} \\   4 \end{array}} \right) \times {(0.297...)^5} \times {(1 - 0.297...)^6}$     (M1)A1

= 0.0586     A1

Note: Allow follow through for incorrect p-values.

[10 marks]

Questions on these discrete distributions have not been generally well answered in the past and it was pleasing to note that many candidates submitted a reasonably good solution to this question. In (b), the determination of the value of p was often successful using a variety of methods including solving the equation $p(1 - p) = {(0.000396{\text{ }} \ldots )^{1/5}}$, graph plotting or using SOLVER on the GDC or even expanding the equation into a ${10^{{\text{th}}}}$ degree polynomial and solving that. Solutions to this particular question exceeded expectations.