(i)     Find ${\rm{P}}(X = 6)$ .

(ii)    Find ${\rm{P}}(X > 6)$ .

(iii)   Find ${\rm{P}}(X = 7|X > 6)$ .

Elena plays a game where she tosses two dice.

If the sum is 6, she wins 3 points.

If the sum is greater than 6, she wins 1 point.

If the sum is less than 6, she loses k points.

Find the value of k for which the game is fair.

(i) number of ways of getting $X = 6$ is 5     A1

${\rm{P}}(X = 6) = \frac{5}{{36}}$     A1     N2

(ii) number of ways of getting $X > 6$ is 21     A1

${\rm{P}}(X > 6) = \frac{{21}}{{36}}\left( { = \frac{7}{{12}}} \right)$     A1     N2

(iii) ${\rm{P}}(X = 7|X > 6) = \frac{6}{{21}}\left( { = \frac{2}{7}} \right)$     A2     N2

[6 marks]

attempt to find ${\rm{P}}(X < 6)$     M1

e.g. $1 - \frac{5}{{36}} - \frac{{21}}{{36}}$

${\rm{P}}(X < 6) = \frac{{10}}{{36}}$     A1

fair game if ${\rm{E}}(W) = 0$ (may be seen anywhere)     R1

attempt to substitute into ${\rm{E}}(X)$ formula     M1

e.g. $3\left( {\frac{5}{{36}}} \right) + 1\left( {\frac{{21}}{{36}}} \right) - k\left( {\frac{{10}}{{36}}} \right)$

correct substitution into ${\rm{E}}(W) = 0$     A1

e.g. $3\left( {\frac{5}{{36}}} \right) + 1\left( {\frac{{21}}{{36}}} \right) - k\left( {\frac{{10}}{{36}}} \right) = 0$

work towards solving     M1

e.g. $15 + 21 - 10k = 0$

$36 = 10k$     A1

$k = \frac{{36}}{{10}}( = 3.6)$     A1     N4

[8 marks]