Find the probability, in terms of $n$, that the game will end on her first draw.

Find the probability, in terms of $n$, that the game will end on her second draw.

third draw.

fourth draw.

Hayley plays the game when $n$ = 5. She pays $20 to play and can earn money back depending on the number of draws it takes to obtain a blue marble. She earns no money back if she obtains a blue marble on her first draw. Let M be the amount of money that she earns back playing the game. This information is shown in the following table.

Find the value of $k$ so that this is a fair game.

$\frac{2}{n}$     A1 N1

[1 mark]

correct probability for one of the draws      A1

eg   P(not blue first) = $\frac{n-2}{n}$,   blue second = $\frac{2}{n-1}$

valid approach      (M1)

eg   recognizing loss on first in order to win on second, P(B' then B),  P(B') × P(B | B'),  tree diagram

correct expression in terms of $n$       A1 N3

eg   $\frac{n-2}{n}\times \frac{2}{n-1}$,  $\frac{2n-4}{n^2-n}$,  $\frac{2\left(n-2\right)}{n\left(n-1\right)}$

[3 marks]

correct working      (A1)

eg   $\frac{3}{5}\times \frac{2}{4}\times \frac{2}{3}$

$\frac{12}{60}\left(=\frac{1}{5}\right)$     A1  N2

[2 marks]

correct working      (A1)

eg  $\frac{3}{5}\times \frac{2}{4}\times \frac{1}{3}\times \frac{2}{2}$

$\frac{6}{60}\left(=\frac{1}{10}\right)$    A1  N2

[2 marks]

correct probabilities (seen anywhere)      (A1)(A1)

eg   $\text{P}\left(\text{1}\right)=\frac{2}{5}$,  $\text{P}\left(\text{2}\right)=\frac{6}{20}$  (may be seen on tree diagram)

valid approach to find E (M) or expected winnings using their probabilities      (M1)

eg   $\text{P}\left(\text{1}\right)\times \left(0\right)+\text{P}\left(\text{2}\right)\times \left(20\right)+\text{P}\left(\text{3}\right)\times \left(8k\right)+\text{P}\left(\text{4}\right)\times \left(12k\right)$,

$\text{P}\left(\text{1}\right)\times \left(-20\right)+\text{P}\left(\text{2}\right)\times \left(0\right)+\text{P}\left(\text{3}\right)\times \left(8k-20\right)+\text{P}\left(\text{4}\right)\times \left(12k-20\right)$

correct working to find E (M) or expected winnings      (A1)

eg   $\frac{2}{5}\left(0\right)+\frac{3}{10}\left(20\right)+\frac{1}{5}\left(8k\right)+\frac{1}{10}\left(12k\right)$,

$\frac{2}{5}\left(-20\right)+\frac{3}{10}\left(0\right)+\frac{1}{5}\left(8k-20\right)+\frac{1}{10}\left(12k-20\right)$

correct equation for fair game      A1

eg   $\frac{3}{10}\left(20\right)+\frac{1}{5}\left(8k\right)+\frac{1}{10}\left(12k\right)=20$,  $\frac{2}{5}\left(-20\right)+\frac{1}{5}\left(8k-20\right)+\frac{1}{10}\left(12k-20\right)=0$

correct working to combine terms in $k$       (A1)

eg   $-8+\frac{14}{5}k-4-2=0$,  $6+\frac{14}{5}k=20$,  $\frac{14}{5}k=14$

$k$ = 5    A1 N0

Note: Do not award the final A1 if the candidate’s FT probabilities do not sum to 1.

[7 marks]