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.