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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]
a.

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]
b.

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

[3]
c.

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]
e.

Markscheme

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

3 4   (0.75, 75%)     (A1)

[1 mark]

a.

3 4 × 1 4 + 1 4 × 3 4   OR   2 × 3 4 × 1 4      (M1)(M1)

Note: Award (M1) for their product  1 4 × 3 4 seen, and (M1) for adding their two products or multiplying their product by 2.

= 3 8 ( 6 16 , 0.375 , 37.5 )      (A1)(ft) (G3)

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

[3 marks]

b.

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

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

[3 marks]

c.

3 4 × 2 5      (M1)

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

= 3 10 ( 0.3 , 30 )      (A1)(ft) (G2)

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

[2 marks]

d.i.

1 3 4 × 2 5 × 3 6   OR  1 4 + 3 4 × 2 5 + 3 4 × 3 5 × 3 6      (M1)(M1)

Note: Award (M1) for 3 4 × 3 5 × 3 6  and (M1) for subtracting their correct probability from 1, or adding to their  1 4 + 3 4 × 2 5 .

= 93 120 ( 31 40 , 0.775 , 77.5 )      (A1)(ft) (G2)

Note: Follow through from their tree diagram.

[3 marks]

d.ii.

3 4 × 3 5 × 3 6 × 120       (M1)(M1)

Note: Award (M1) for  3 4 × 3 5 × 3 6 ( 3 4 × 3 5 × 3 6 OR 27 120 OR 9 40 )  and (M1) for multiplying by 120.

= 27      (A1)(ft) (G3)

Note: Follow through from their tree diagram or their  3 4 × 3 5 × 3 6  from their calculation in part (d)(ii).

[3 marks]

e.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.
[N/A]
d.i.
[N/A]
d.ii.
[N/A]
e.

Syllabus sections

Topic 4—Statistics and probability » SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams
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Topic 4—Statistics and probability

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