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Date	May 2021	Marks available	2	Reference code	21M.1.SL.TZ1.10
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Complete	Question number	10	Adapted from	N/A

Question

A game is played where two unbiased dice are rolled and the score in the game is the greater of the two numbers shown. If the two numbers are the same, then the score in the game is the number shown on one of the dice. A diagram showing the possible outcomes is given below.

Let $T$ $T$ be the random variable “the score in a game”.

Find the probability that

Complete the table to show the probability distribution of $T$ $T$ .

[2]

a.

a player scores at least $3$ $3$ in a game.

[1]

b.i.

a player scores $6$ $6$ , given that they scored at least $3$ $3$ .

[2]

b.ii.

Find the expected score of a game.

[2]

c.

Markscheme

A2

Note: Award A1 if three to five probabilities are correct.

[2 marks]

a.

$\frac{32}{36} (\frac{8}{9}, 0.888888 \dots, 88.9 %)$ $\frac{32}{36} (\frac{8}{9}, 0.888888 \dots, 88.9 %)$ (A1)

[1 mark]

b.i.

use of conditional probability (M1)

e.g. denominator of $32$ $32$ OR denominator of $0.888888 \dots$ $0.888888 \dots$ , etc.

$\frac{11}{32} (0.34375, 34.4 %)$ $\frac{11}{32} (0.34375, 34.4 %)$ A1

[2 marks]

b.ii.

$\frac{1 \times 1 + 3 \times 2 + 5 \times 3 + \dots + 11 \times 6}{36}$ $\frac{1 \times 1 + 3 \times 2 + 5 \times 3 + \dots + 11 \times 6}{36}$ (M1)

$= \frac{161}{36} (4 \frac{17}{36}, 4.47, 4.47222 \dots)$ $= \frac{161}{36} (4 \frac{17}{36}, 4.47, 4.47222 \dots)$ A1

[2 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.

Syllabus sections

Topic 4—Statistics and probability » SL 4.7—Discrete random variables

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17M.1.AHL.TZ1.H_10a:
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17M.1.AHL.TZ1.H_10b.i:
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18M.2.AHL.TZ1.H_10b:
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SPM.1.SL.TZ0.12a:
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18M.2.SL.TZ1.S_2c:
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19M.2.SL.TZ1.S_10c.ii:
Hence, find the value of $y$ $y$ .
17N.2.SL.TZ0.S_4b:
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17M.2.AHL.TZ2.H_10c:
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16N.2.AHL.TZ0.H_1a:
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17M.2.SL.TZ1.S_4b.ii:
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18N.1.SL.TZ0.S_9c:
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17M.2.AHL.TZ2.H_10e:
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19M.2.SL.TZ1.S_10a.ii:
Find the probability of rolling two or more red faces.
19M.2.SL.TZ1.S_10a.i:
Find the probability of rolling exactly one red face.
17M.1.AHL.TZ1.H_10b.ii:
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17M.2.SL.TZ1.S_4b.i:
Hence, find the probability that exactly $k$ students are left handed;
16N.1.AHL.TZ0.H_2a:
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16N.1.AHL.TZ0.H_1:
Find the coordinates of the point of intersection of the planes defined by the equations $x + y + z = 3,{\text{ }}x - y + z = 5$ and $x + y + 2z = 6$ .
17M.2.AHL.TZ2.H_10b:
Find ${\text{E}}(X)$ .
17N.2.SL.TZ0.S_4a:
Find the value of $k$ .
16N.1.AHL.TZ0.H_2b:
Find the expected value of $X$ .
18N.1.SL.TZ0.S_9a.i:
Find the probability, in terms of $n$ , that the game will end on her first draw.
18M.2.SL.TZ1.S_2a:
Find the value of k.
16N.2.AHL.TZ0.H_1b:
Find the value of ${\text{Var}}(X)$ .
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Find the probability that on a randomly selected day, Steffi does not visit Will’s house.
17M.2.AHL.TZ2.H_1a:
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18N.1.SL.TZ0.S_9b.ii:
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18M.1.AHL.TZ1.H_3b:
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16N.2.SL.TZ0.S_7a:
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20N.2.SL.TZ0.S_3a:
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17M.2.SL.TZ2.S_10d:
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EXN.1.SL.TZ0.12b:
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19M.2.SL.TZ1.S_10b:
Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is ${\frac{1}{3}}$ .
17N.2.SL.TZ0.S_4c:
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Find $q$ .
19M.2.AHL.TZ2.H_10d:
In any given year of 365 days, the probability that Steffi does not visit Will for at most $n$ days in total is 0.5 (to one decimal place). Find the value of $n$ .
18M.2.SL.TZ2.S_10b:
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17M.2.SL.TZ2.S_B10c:
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18N.1.SL.TZ0.S_9b.i:
third draw.
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21M.1.SL.TZ1.10b.i:
a player scores at least $3$ in a game.
21M.1.SL.TZ1.10b.ii:
a player scores $6$ , given that they scored at least $3$ .
18M.2.AHL.TZ1.H_10c:
Given that ${\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)$ , and that 0.25 < s < 0.4 , find the value of s.
21M.1.SL.TZ1.10c:
Find the expected score of a game.
18M.2.SL.TZ2.S_10c:
Find the probability that a bag of apples selected at random contains at most one small apple.
17M.2.SL.TZ2.S_10a.ii:
Find $p$ .
18M.2.SL.TZ1.S_2b:
Calculate the expected value of the score.
18M.2.SL.TZ2.S_10a.i:
Write down the value of k.
17M.2.AHL.TZ2.H_10a:
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17M.2.AHL.TZ2.H_1b:
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19M.2.AHL.TZ2.H_10c:
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17M.2.SL.TZ2.S_10b.iii:
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Find the probability, in terms of $n$ , that the game will end on her second draw.
17M.1.AHL.TZ1.H_10b.iii:
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19M.2.SL.TZ1.S_10d:
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.
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EXN.1.SL.TZ0.12c:
A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $E (X)$ .
17M.1.AHL.TZ1.H_10c.ii:
Hence state the interquartile range of $X$ .

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Topic 4—Statistics and probability