(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).