Over a one month period, Ava and Sven play a total of n games of tennis.

The probability that Ava wins any game is 0.4. The result of each game played is independent of any other game played.

Let X denote the number of games won by Ava over a one month period.

(a)     Find an expression for P(X = 2) in terms of n.

(b)     If the probability that Ava wins two games is 0.121 correct to three decimal places, find the value of n.

(a)     \(X \sim {\text{B}}(n,{\text{ }}0.4)\)     (A1)

Using \({\text{P}}(X = x) = \left( {\begin{array}{*{20}{c}}
  n \\
  r
\end{array}} \right){(0.4)^x}{(0.6)^{n - x}}\)     (M1)

\({\text{P}}(X = 2) = \left( {\begin{array}{*{20}{c}}
  n \\
  2
\end{array}} \right){(0.4)^2}{(0.6)^{n - 2}}\)     \(\left( { = \frac{{n(n - 1)}}{2}{{(0.4)}^2}{{(0.6)}^{n - 2}}} \right)\)     A1     N3

(b)     P(X = 2) = 0.121     A1

Using an appropriate method (including trial and error) to solve their equation.     (M1)

n = 10     A1     N2

Note: Do not award the last A1 if any other solution is given in their final answer.

[6 marks]

Part (a) was generally well done. The most common error was to omit the binomial coefficient i.e. not identifying that the situation is described by a binomial distribution.

Finding the correct value of n in part (b) proved to be more elusive. A significant proportion of candidates attempted algebraic approaches and seemingly did not realise that the equation could only be solved numerically. Candidates who obtained n = 10 often accomplished this by firstly attempting to solve the equation algebraically before ‘resorting’ to a GDC approach. Some candidates did not specify their final answer as an integer while others stated n = 1.76 as their final answer.