Discrete random variable questions are found in paper 2, section B (Statistics section) of the H2 A Level Math exam. In this post, you will find the worked solutions for past A Level questions on discrete random variable.
A list of past questions from each statistics topic can be found here.
A quick summary of formulae in each statistics topics can be found here.
Past A Level Discrete Random Variable Questions
These are the past A Level discrete random variable questions for the new syllabus (syllabus 9758) which started from 2017.
Click on the link to go straight to the worked solutions for each question.
Discrete Random Variable Question (DRV): 2017 Paper 2 Question 5
(i) P(T=2) = (6/9)(5/8) = 5/12
P(T=3) = (6/9)(5/8)(3/7)(2) = 5/14
P(T=4) = (6/9)(5/8)(3/7)(2/6)(3)= 5/28
P(T=5) = (6/9)(5/8)(3/7)(2/6)(1/5)(4)= 1/21
(ii) E(T) = 5/12(2) + 5/14(3) + (5/28)(4)+(1/21)(5) = 20/7
E(T2) = 5/12(22) + 5/14(32) + (5/28)(42)+(1/21)(52) = 125/14
Var(T) = 125/14 – (20/7)2 = 75/98
(iii) P(T≥4) = 5/28 + 1/21 = 19/84
Let X be the number of games played out of 15 where Lee takes at least 5 counters out of the bag in each game.
X~B(15, 19/84)
P(X≥5) =1 -P(X≤4) = 0.238 ( 3 s.f.)
Discrete Random Variable Question (DRV): 2018 Paper 2 Question 8
(i) two 3, three 4, n 5.
Possible values of S:
| ball 1 /ball 2 | 3 | 4 | 5 |
| 3 | 6 | 7 | 8 |
| 4 | 7 | 8 | 9 |
| 5 | 8 | 9 | 10 |
| S | 6 | 7 | 8 | 9 | 10 |
| P(S=s) | 2/ [(n+4)(n+5)] | 12/ [(n+4)(n+5)] | (6+4n)/[(n+4)(n+5)] | 6n/[(n+4)(n+5)] | n(n-1)/[(n+4)(n+5)] |
(ii) when n = 1, P(S=10) = 0
There is no possibility that S = 10. In order for S=10, two balls numbered 5 have to be picked. However, since there is only 1 ball numbered 5, it is not possible to pick 2 balls numbered 5 (without replacement).
(iii) E(S) = 12/ [(n+4)(n+5)] +84/ [(n+4)(n+5)] +(48+32n)/[(n+4)(n+5)] + 54n/[(n+4)(n+5)] +10n/(n-1)/[(n+4)(n+5)] =(144+76n+10n2)/[(n+4)(n+5)] =2(n+4)(5n+18)/[(n+4)(n+5)] = (10n+36)/[(n+5)]
E(S2) = 72/ [(n+4)(n+5)] +588/ [(n+4)(n+5)] +(384+256n)/[(n+4)(n+5)] + 486n/[(n+4)(n+5)] +100n/(n-1)/[(n+4)(n+5)] =(1044+642n+100n2)/[(n+4)(n+5)]
Var(S) =E(S2) – [E(S)]2 = (1044+642n+100n2)/[(n+4)(n+5)] – (10n+36)2/[(n+5)2]
= [(1044+642n+100n2)(n+5) -(100n2+720n+1296)(n+4)]/ [(n+4)(n+5)2]
= [22n2 + 78n + 36]/ [(n+4)(n+5)2]
g(n) = 22n2 + 78n + 36
Discrete Random Variable Question (DRV): 2020 Paper 2 Question 5
(i) 1 green; r red; 2r blue; Total number of discs = 3r+1
green – 0 points; red – 5 points; blue – 2 points
possible scores
| green (0) | red (5) | blue (2) | |
| green (0) | N/A [not possible, only 1 green] | 0 | 0 |
| red (5) | 0 | 25 | 10 |
| blue (2) | 0 | 10 | 4 |
Tina’s possible scores are: 0, 4, 10, 25
(ii) Let Tina’s score be S.
| S | 0 | 4 | 10 | 25 |
| P(S=s) | 2(3r)/[3r(3r+1)] = 2/(3r+1) | 2r(2r-1)/[3r(3r+1)] =(4r-2)/[3(3r+1)] | 4r2/[3r(3r+1)] =4r/[3(3r+1)] | r(r-1)/[3r(3r+1)] =(r-1)/[3(3r+1)] |
E(S) = 0+ (16r-8)/[3(3r+1)] +40r/[3(3r+1)] + (25r-25)/[3(3r+1)] =(81r-33)/[3(3r+1)]=(27r-11)/(3r+1)
E(S2) = (64r-32)/[3(3r+1)] + 400r/[3(3r+1)] +(625r-625)/[3(3r+1)] =(1089r – 657)/ [3(3r+1)] = (363r-219)/(3r+1)
Var(S) =(363r-219)/(3r+1) – [ (27r-11)/(3r+1) ] 2 = [(363r-219)(3r+1) – (27r-11)2]/ (3r+1)2 = (360r2 + 300r-340 )/(3r+1)2
(iii)when Var(S) = 38,
(360r2 + 300r-340 )/(3r+1)2 = 38
Using Graphic calculator, and rejecting negative values of r,
r = 3
Discrete Random Variable Question (DRV): 2021 Paper 2 Question 6
Given:
| score | 1 | 2 | 3 | 4 | 5 |
| probability | 0.2 | 0.3 | p | p | q |
Also given variance = 1.61
0.2 + 0.3 + p + p + q = 1
2p + q = 0.5
q= 0.5 – 2p —-(1)
Let S be the score.
E(S) = 0.2 + 0.6 + 3p + 4p + 5q = 0.8 + 7p + 5q sub (1) in
E(S) = 0.8 + 7p +5(0.5-2p) = 3.3 – 3p
E(S2) = 0.2 + 1.2 + 9p + 16p + 25q = 1.4 + 25p + 25q sub (1) in
E(S2) = 1.4 + 25p + 25(0.5 – 2p) = 13.9 – 25p
Var(S) = E(S2) -[E(S2)] = 13.9 – 25p -(3.3-3p)2 = 13.9 – 25p – 10.89 +19.8p – 9p2 = 3.01-5.2p – 9p2
3.01 – 5.2p- 9p2 =1.61
By Graphic Calculator, and rejecting negative values
p = 0.2
E(S) = 3.3 – 3(0.2) = 2.7
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