Find, in terms of $p$, an expression for ${\text{P}}(X = 4)$.

(i)     Determine the value of $p$ for which ${\text{P}}(X = 4)$ is a maximum.

(ii)     For this value of $p$, determine the expected number of heads.

$X \sim {\text{B}}(5,{\text{ }}p)$    (M1)

${\text{P}}(X = 4) = \left( {\begin{array}{*{20}{c}} 5 \\ 4 \end{array}} \right){p^4}(1 - p)$ (or equivalent)     A1

[2 marks]

(i)     $\frac{{\text{d}}}{{{\text{d}}p}}(5{p^4} - 5{p^5}) = 20{p^3} - 25{p^4}$     M1A1

$5{p^3}(4 - 5p) = 0 \Rightarrow p = \frac{4}{5}$    M1A1

Note:     Do not award the final A1 if $p = 0$ is included in the answer.

(ii)     ${\text{E}}(X) = np = 5\left( {\frac{4}{5}} \right)$     (M1)

$= 4$    A1

[6 marks]

This question was generally very well done and posed few problems except for the weakest candidates.

This question was generally very well done and posed few problems except for the weakest candidates.