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Date May 2019 Marks available 5 Reference code 19M.2.SL.TZ1.S_10
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Show that Question number S_10 Adapted from N/A

Question

There are three fair six-sided dice. Each die has two green faces, two yellow faces and two red faces.

All three dice are rolled.

Ted plays a game using these dice. The rules are:

The random variable D  ($) represents how much is added to his winnings after a turn.

The following table shows the distribution for D , where $ w represents his winnings in the game so far.

Find the probability of rolling exactly one red face.

[2]
a.i.

Find the probability of rolling two or more red faces.

[3]
a.ii.

Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is  1 3 .

[5]
b.

Write down the value of x .

[1]
c.i.

Hence, find the value of y .

[2]
c.ii.

Ted will always have another turn if he expects an increase to his winnings.

Find the least value of w for which Ted should end the game instead of having another turn.

[3]
d.

Markscheme

valid approach to find P(one red)     (M1)

eg   n C a × p a × q n a ,   B ( n p ) ,   3 ( 1 3 ) ( 2 3 ) 2 ,   ( 3 1 )

listing all possible cases for exactly one red (may be indicated on tree diagram)

P(1 red) = 0.444  ( = 4 9 )    [0.444, 0.445]           A1  N2

 [3 marks] [5 maximum for parts (a.i) and (a.ii)]

a.i.

valid approach     (M1)

eg  P( X = 2 ) + P( X = 3 ), 1 − P( X  ≤ 1),  binomcdf ( 3 , 1 3 , 2 , 3 )

correct working       (A1)

eg    2 9 + 1 27 ,   0.222 + 0.037 ,   1 ( 2 3 ) 3 4 9

0.259259

P(at least two red) = 0.259  ( = 7 27 )           A1  N3

[3 marks]  [5 maximum for parts (a.i) and (a.ii)]

a.ii.

recognition that winning $10 means rolling exactly one green        (M1)

recognition that winning $10 also means rolling at most 1 red        (M1)

eg “cannot have 2 or more reds”

correct approach        A1

eg  P(1G ∩ 0R) + P(1G ∩ 1R),  P(1G) − P(1G ∩ 2R),

      “one green and two yellows or one of each colour”

Note: Because this is a “show that” question, do not award this A1 for purely numerical expressions.

one correct probability for their approach        (A1)

eg    3 ( 1 3 ) ( 1 3 ) 2 ,   6 27 3 ( 1 3 ) ( 2 3 ) 2 1 9 ,   2 9

correct working leading to 1 3       A1

eg    3 27 + 6 27 12 27 3 27 ,   1 9 + 2 9

probability =  1 3       AG N0

[5 marks]

b.

x = 7 27 ,  0.259 (check FT from (a)(ii))      A1 N1

[1 mark]

c.i.

evidence of summing probabilities to 1       (M1)

eg    = 1 ,   x + y + 1 3 + 2 9 + 1 27 = 1 ,   1 7 27 9 27 6 27 1 27

0.148147  (0.148407 if working with their x value to 3 sf)

y = 4 27   (exact), 0.148     A1 N2

[2 marks]

c.ii.

correct substitution into the formula for expected value      (A1)

eg   w 7 27 + 10 9 27 + 20 6 27 + 30 1 27

correct critical value (accept inequality)       A1

eg    w = 34.2857  ( = 240 7 ) w  > 34.2857

$40      A1 N2

[3 marks]

d.

Examiners report

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

Syllabus sections

Topic 4—Statistics and probability » SL 4.5—Probability concepts, expected numbers
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Topic 4—Statistics and probability

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