In a game there are n players, where \(n > 2\) . Each player has a disc, one side of which is red and one side blue. When thrown, the disc is equally likely to show red or blue. All players throw their discs simultaneously. A player wins if his disc shows a different colour from all the other discs. Players throw repeatedly until one player wins.

Let X be the number of throws each player makes, up to and including the one on which the game is won.

(a)     State the distribution of X .

(b)     Find \({\text{P}}(X = x)\) in terms of n and x .

(c)     Find \({\text{E}}(X)\) in terms of n .

(d)     Given that n = 7 , find the least number, k , such that \({\text{P}}(X \leqslant k) > 0.5\) .

(a)     geometric distribution     A1

[1 mark]

 

(b)     let R be the event throwing the disc and it landing on red and 

let B be the event throwing the disc and it landing on blue 

\({\text{P}}(X = 1) = p = {\text{P}}\left( {1B{\text{ and }}(n - 1)R{\text{ or }}1R{\text{ and }}(n - 1)B} \right)\)     (M1)

\( = n \times \frac{1}{2} \times {\left( {\frac{1}{2}} \right)^{n - 1}} + n \times \frac{1}{2} \times {\left( {\frac{1}{2}} \right)^{n - 1}}\)     (A1)

\( = \frac{n}{{{2^{n - 1}}}}\)     A1

hence \({\text{P}}(X = x) = \frac{n}{{{2^{n - 1}}}}{\left( {1 - \frac{n}{{{2^{n - 1}}}}} \right)^{x - 1}},{\text{ }}(x \geqslant 1)\)     A1

Notes: \(x \geqslant 1\) not required for final A1.

Allow FT for final A1.

 

[4 marks]

 

(c)     \({\text{E}}(X) = \frac{1}{p}\)

\( = \frac{{{2^{n - 1}}}}{n}\)     A1

[1 mark]

 

(d)     when \(n = 7\) , \({\text{P}}(X = x) = {\left( {1 - \frac{7}{{64}}} \right)^{x - 1}} \times \frac{7}{{64}}\)     (M1)

\( = \frac{7}{{64}} \times {\left( {\frac{{57}}{{64}}} \right)^{x - 1}}\)

\({\text{P}}(X \leqslant k) = \sum\limits_{x = 1}^k {\frac{7}{{64}} \times {{\left( {\frac{{57}}{{64}}} \right)}^{x - 1}}} \)     (M1)(A1)

\( \Rightarrow \frac{7}{{64}} \times \frac{{1 - {{\left( {\frac{{57}}{{64}}} \right)}^k}}}{{1 - \frac{{57}}{{64}}}} > 0.5\)     (M1)(A1)

\( \Rightarrow 1 - {\left( {\frac{{57}}{{64}}} \right)^k} > 0.5\)

\( \Rightarrow {\left( {\frac{{57}}{{64}}} \right)^k} < 0.5\)

\( \Rightarrow k > \frac{{\log 0.5}}{{\log \frac{{57}}{{64}}}}\)     (M1)

\( \Rightarrow k > 5.98\)     (A1)

\( \Rightarrow k = 6\)     A1

Note: Tabular and other GDC methods are acceptable.

 

[8 marks]

Total [14 marks]

This question was found difficult by the majority of candidates and few fully correct answers were seen. Few candidates were able to find \({\text{P}}(X = x)\) in terms of n and x and many did not realise that the last part of the question required them to find the sum of a series. However, better candidates received over 75% of the marks because the answers could be followed through.