Draw a Venn diagram to represent the given information, using sets labelled \(B\), \(C\) and \(H\).

Show that \(x = 3\).

Write down the value of \(n(B \cap C)\).

Find the probability that this person

(i)     went on at most one trip;

(ii)     went on the coach trip, given that this person also went on both the helicopter trip and the boat trip.

     (A5)

Notes:     Award (A1) for rectangle and three labelled intersecting circles (U need not be seen),

(A1) for 3 in the correct region,

(A1) for 8 in the correct region,

(A1) for 5, 13 and 16 in the correct regions,

(A1) for \(x\), \(2x\) and \(4x\) in the correct regions.

[5 marks]

\(8 + 13 + 16 + 3 + 5 + x + 2x + 4x = 66\)    (M1)

Note:     Award (M1) for either a completely correct equation or adding all the terms from their diagram in part (a) and equating to 66.

Award (M0)(A0) if their equation has no \(x\).

\(7x = 66 - 45\) OR \(7x + 45 = 66\)     (A1)

Note:     Award (A1) for adding their like terms correctly, but only when the solution to their equation is equal to 3 and is consistent with their original equation.

\(x = 3\)    (AG)

Note:     The conclusion \(x = 3\) must be seen for the (A1) to be awarded.

[2 marks]

15     (A1)(ft)

Note:     Follow through from part (a). The answer must be an integer.

[1 mark]

(i)     \(\frac{{42}}{{66}}{\text{ }}\left( {\frac{7}{{11}},{\text{ }}0.636,{\text{ }}63.6\% } \right)\)     (A1)(ft)(A1)(G2)

Note:     Award (A1)(ft) for numerator, (A1) for denominator. Follow through from their Venn diagram.

(ii)     \(\frac{3}{9}{\text{ }}\left( {\frac{1}{3},{\text{ }}0.333,{\text{ }}33.3\% } \right)\)     (A1)(A1)(ft)(G2)

Note:     Award (A1) for numerator, (A1)(ft) for denominator. Follow through from their Venn diagram.

[4 marks]