(a)     The heating in a residential school is to be increased on the third frosty day during the term. If the probability that a day will be frosty is 0.09, what is the probability that the heating is increased on the \({25^{{\text{th}}}}\) day of the term?

(b)     On which day is the heating most likely to be increased?

(a)     the distribution is NB(3, 0.09)     (M1)(A1)

the probability is \(\left( {\begin{array}{*{20}{c}}
  {24} \\
  2
\end{array}} \right){0.91^{22}} \times {0.09^3} = 0.0253\)     (M1)(A1)A1

[5 marks]

(b)     P(Heating increased on \({n^{{\text{th}}}}\) day)

\(\left( {\begin{array}{*{20}{c}}
  {n - 1} \\
  2
\end{array}} \right){0.91^{n - 3}} \times {0.09^3}\)     (M1)(A1)(A1)

by trial and error n = 23 gives the maximum probability     (M1)A3

(neighbouring values: 0.02551 (n = 22) ; 0.02554 (n = 23) ; 0.02545 (n = 24) )

[7 marks]

Total [12 marks]

Most candidates understood the context of this question, and the negative binomial distribution was usually applied, albeit occasionally with incorrect parameters. Good solutions were seen to part(b), using lists in their GDC or trial and error.