Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js

User interface language: English | Español

Date May 2019 Marks available 1 Reference code 19M.2.SL.TZ1.S_10
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Write down 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 13.

[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  nCa×pa×qna,  B(np),  3(13)(23)2,  (31)

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

P(1 red) = 0.444 (=49)   [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(3,13,2,3)

correct working       (A1)

eg   29+127,   0.222 + 0.037 ,  1(23)349

0.259259

P(at least two red) = 0.259 (=727)          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(13)(13)2,  6273(13)(23)219,  29

correct working leading to 13      A1

eg   327+6271227327,  19+29

probability = 13      AG N0

[5 marks]

b.

x=727,  0.259 (check FT from (a)(ii))      A1 N1

[1 mark]

c.i.

evidence of summing probabilities to 1       (M1)

eg   =1,  x+y+13+29+127=1,  1727927627127

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

y=427  (exact), 0.148     A1 N2

[2 marks]

c.ii.

correct substitution into the formula for expected value      (A1)

eg   w727+10927+20627+30127

correct critical value (accept inequality)       A1

eg   w = 34.2857  (=2407)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
Show 158 related questions
Topic 4—Statistics and probability » SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams
Topic 4—Statistics and probability » SL 4.7—Discrete random variables
Topic 4—Statistics and probability » SL 4.8—Binomial distribution
Topic 4—Statistics and probability

View options