(i)     Write down the mode of $X$ .

(ii)     Find the exact value of $p$ if ${\rm{Var}}(X) = \frac{{28}}{9}$ .

(i)     Find the smallest value of $n$ for which the probability of Arthur walking to school on the next $n$ days is less than $0.01$.

(ii)     Find the probability that Arthur cycles to school for the third time on the last of eight successive days.

(i)     the mode is $1$     A1

(ii)     attempt to solve $\frac{{1 - p}}{{{p^2}}} = \frac{{28}}{9}$     M1

obtain $p = \frac{3}{7}$     A1

Note: $p = 0.429$ is awarded M1A0.

[3 marks]

(i)     require least $n$ such that

${0.55^n} < 0.01$     (M1)

EITHER

listing values: $0.55$, $0.3025$, $0.166$, $0.091$, $0.050$, $0.028$, $0.015$, $0.0084$     (M1)

obtain $n = 8$     A1

OR

$n > \frac{{\ln 0.01}}{{\ln 0.55}} = 7.70 \ldots$     (M1)

obtain $n = 8$     A1

(ii)     recognition of negative binomial     (M1)

$X \sim NB(3,0.45)$

${\rm{P}}(X = 8) = \left( \begin{array}{l} 7\\ 2 \end{array} \right) \times {0.45^3} \times {0.55^5}$     (A1)

$= 0.0963$     A1

Note: If $0.45$ and $0.55$ are mixed up, count it as a misread – probability in that case is $0.0645$.

[6 marks]

(a)(i) A surprising number of candidates were unaware of the definition of the mode of a distribution.

(a)(ii) Generally well done, although a few candidates gave a decimal answer.

(b) Generally well done, and it was pleasing that most were familiar with the direct use of the negative binomial distribution in (ii).