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Date	May 2018	Marks available	3	Reference code	18M.2.SL.TZ1.T_5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	T_5	Adapted from	N/A

Question

Contestants in a TV gameshow try to get through three walls by passing through doors without falling into a trap. Contestants choose doors at random.
If they avoid a trap they progress to the next wall.
If a contestant falls into a trap they exit the game before the next contestant plays.
Contestants are not allowed to watch each other attempt the game.

The first wall has four doors with a trap behind one door.

Ayako is a contestant.

Natsuko is the second contestant.

The second wall has five doors with a trap behind two of the doors.

The third wall has six doors with a trap behind three of the doors.

The following diagram shows the branches of a probability tree diagram for a contestant in the game.

Write down the probability that Ayako avoids the trap in this wall.

[1]

Find the probability that only one of Ayako and Natsuko falls into a trap while attempting to pass through a door in the first wall.

[3]

Copy the probability tree diagram and write down the relevant probabilities along the branches.

[3]

A contestant is chosen at random. Find the probability that this contestant fell into a trap while attempting to pass through a door in the second wall.

[2]

d.i.

A contestant is chosen at random. Find the probability that this contestant fell into a trap.

[3]

d.ii.

120 contestants attempted this game.

Find the expected number of contestants who fell into a trap while attempting to pass through a door in the third wall.

[3]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\frac{3}{4}$ (0.75, 75%) (A1)

[1 mark]

$\frac{3}{4} \times \frac{1}{4} + \frac{1}{4} \times \frac{3}{4}$ OR $2 \times \frac{3}{4} \times \frac{1}{4}$ (M1)(M1)

Note: Award (M1) for their product $\frac{1}{4} \times \frac{3}{4}$ seen, and (M1) for adding their two products or multiplying their product by 2.

$= \frac{3}{8}\,\,\,\,\left( {\frac{6}{{16}},\,\,0.375,\,\,37.5{\text{% }}} \right)$ (A1)(ft) (G3)

Note: Follow through from part (a), but only if the sum of their two fractions is 1.

[3 marks]

(A1)(ft)(A1)(A1)

Note: Award (A1) for each correct pair of branches. Follow through from part (a).

[3 marks]

$\frac{3}{4} \times \frac{2}{5}$ (M1)

Note: Award (M1) for correct probabilities multiplied together.

$= \frac{3}{{10}}\,\,\,\left( {0.3,\,\,30{\text{% }}} \right)$ (A1)(ft) (G2)

Note: Follow through from their tree diagram or part (a).

[2 marks]

d.i.

$1 - \frac{3}{4} \times \frac{2}{5} \times \frac{3}{6}$ OR $\frac{1}{4} + \frac{3}{4} \times \frac{2}{5} + \frac{3}{4} \times \frac{3}{5} \times \frac{3}{6}$ (M1)(M1)

Note: Award (M1) for $\frac{3}{4} \times \frac{3}{5} \times \frac{3}{6}$ and (M1) for subtracting their correct probability from 1, or adding to their $\frac{1}{4} + \frac{3}{4} \times \frac{2}{5}$ .

$= \frac{{93}}{{120}}\,\,\,\,\left( {\frac{{31}}{{40}},\,\,0.775,\,\,77.5{\text{% }}} \right)$ (A1)(ft) (G2)

Note: Follow through from their tree diagram.

[3 marks]

d.ii.

$\frac{3}{4} \times \frac{3}{5} \times \frac{3}{6} \times 120$ (M1)(M1)

Note: Award (M1) for $\frac{3}{4} \times \frac{3}{5} \times \frac{3}{6}\,\,\,\,\left( {\frac{3}{4} \times \frac{3}{5} \times \frac{3}{6}\,\,{\text{OR}}\,\,\frac{{27}}{{120}}\,\,{\text{OR}}\,\,\frac{9}{{40}}} \right)$ and (M1) for multiplying by 120.