Download 7D SA Review Probability Markscheme

SA Review Probability
1.
(b)
(i)
A1A1A1
N3
Note: Award A1 for each complementary pair of
probabilities,
4
2 3
2 4
1
i.e. and , and , and .
6
6 5
5 5
5
(ii)
P(Y = 0) =
2 1 2
 
5 5 30
A1
 4 2 2 4
P(Y = 1) = P(RG) + P(GR)      
 6 5 6 5
16
=
30
4 3 12
P(Y = 2) =  
6 5 30
For forming a distribution
y
P(Y = y)
0
2
30
1
16
30
M1
A1
(A1)
M1
2
12
30
N4
(c)
(d)
2  1
 
6  3
4  2
P(Bag B) =
 
6  3
For summing P(A ∩ RR) and P(B ∩ RR)
1 1 2 12
Substituting correctly P(RR) =   
3 10 3 30
= 0.3
P(Bag A) =
For recognising that P(1 or 6 │ RR) = P(A│RR) =
1 27

30 90
= 0.111
=
(A1)
(A1)
(M1)
A1
A1
P( A  RR)
P( RR)
N3
(M1)
A1
A1
N2
[16]
2.
(a)
E(X) = 2
(b)
evidence of appropriate approach involving binomial
10 
e.g.   (0.2)3, (0.2)3(0.8)7, X ~ B(10, 0.2)
3
P(X = 3) = 0.201
(c)
A1
N1
(M1)
A1
METHOD 1
P(X ≤ 3) = 0.10737 + 0.26844 + 0.30199 + 0.20133 (= 0.87912...)
evidence of using the complement (seen anywhere)
e.g. 1 – any probability, P(X > 3) = 1 – P(X ≤ 3)
P(X > 3) = 0.121
N2
(A1)
(M1)
A1
N2
METHOD 2
recognizing that P(X > 3) = P(X ≥ 4)
(M1)
e.g. summing probabilities from X = 4 to X = 10
correct expression or values
(A1)
10
10 
  (0.2)10–r(0.8)r
e.g.
r 4  r 
0.08808 + 0.02642 + 0.005505 + 0.000786 + 0.0000737 + 0.000004 + 0.0000001
P(X > 3) = 0.121
A1
N2

[6]
3.
(a)
(b)
(c)
evidence of binomial distribution (may be seen in parts (b) or (c))
e.g. np, 100  0.04
mean = 4
100 
 0.04 6 0.96 94
P(X = 6) = 
6


= 0.105
for evidence of appropriate approach
e.g. complement, 1  P(X = 0)
P(X = 0) = (0.96)100 = 0.01687...
P(X  1) = 0.983
(M1)
A1
N2
(A1)
A1
N2
(M1)
(A1)
A1
N2
[7]
4.
(a)
(b)
(i)
s=1
A1
(ii)
evidence of appropriate approach
e.g. 21–16, 12 + 8 – q =15
q=5
(iii)
p = 7, r = 3
(i)
P(art|music) =
(ii)
METHOD 1
12  3 
Part    
16  4 
evidence of correct reasoning
3 5
e.g. 
4 8
5
8
N1
(M1)
A1
N2
A1A1
N2
A2
N2
A1
R1
5
the events are not independent
AG
N0
METHOD 2
96  3 
 
256  8 
evidence of correct reasoning
12 8
5
e.g.
 
16 16 16
the events are not independent
P(art) × P(music) =
(c)
3
= (seen anywhere)
16
7
P(second takes only art)= (seen anywhere)
15
evidence of valid approach
3 7
e.g.

16 15
21  7 
P(music and art)=
 
240  80 
P(first takes only music) =
A1
R1
AG
N0
4
N2
4
A1
A1
(M1)
A1
[13]
5.
(a)
(b)
evidence of valid approach involving A and B
e.g. P(A ∩ pass) + P(B ∩ pass), tree diagram
correct expression
e.g. P(pass) = 0.6 × 0.8 + 0.4 × 0.9
P(pass) = 0.84
(M1)
(A1)
A1
N2
3
evidence of recognizing complement (seen anywhere)
(M1)
e.g. P(B) = x, P(A) = 1 – x, 1 – P(B), 100 – x, x + y =1
evidence of valid approach
(M1)
e.g. 0.8(1 – x) + 0.9x, 0.8x + 0.9y
correct expression
A1
e.g. 0.87 = 0.8(1 – x) + 0.9x, 0.8 × 0.3 + 0.9 × 0.7 = 0.87, 0.8x + 0.9y = 0.87
70 % from B
A1
N2
4
[7]
6.
4
5
(a)
p=
(b)
multiplying along the branches
1 1 12
e.g.  ,
5 4 40
adding products of probabilities of two mutually exclusive paths
1 1 4 3 1 12
e.g.    , 
5 4 5 8 20 40
14  7 
P(B) =
 
40  20 
(M1)
appropriate approach which must include A′ (may be seen on diagram)
P( A  B ) 
P( A  B) 
 do not accept

e.g.
P( B ) 
P( B) 
(M1)
(c)
A1
N1
(M1)
A1
N2
4 3

P(A′│B) = 5 8
7
20
12  6 
P(A′│B) =
 
14  7 
(A1)
A1
N2
[7]
7.
(a)
(i)
(ii)
(b)
(i)
(ii)
(c)
valid approach
1
e.g. np, 5 ×
5
E(X) = 1
(M1)
evidence of appropriate approach involving binomial
 1
e.g. X ~ B  5, 
 5
recognizing that Mark needs to answer 3 or more questions correctly
e.g. P(X ≥ 3)
valid approach
e.g. 1 – P(X ≤ 2), P(X = 3) + P(X =4) + P(X = 5)
P(pass) = 0.0579
(M1)
evidence of summing probabilities to 1
e.g. 0.67 + 0.05 + (a + 2b) + ... + 0.04 = 1
some simplification that clearly leads to required answer
e.g. 0.76 + 4a + 2b = 1
4a + 2b = 0.24
(M1)
correct substitution into the formula for expected value
e.g. 0(0.67) + 1(0.05) + ... + 5(0.04)
some simplification
e.g. 0.05 + 2a + 4b + ... + 5(0.04) = 1
correct equation
e.g. 13a + 5b = 0.75
evidence of solving
a = 0.05, b = 0.02
(A1)
attempt to find probability Bill passes
e.g. P(Y ≥ 3)
correct value 0.19
Bill (is more likely to pass)
A1
N2
(A1)
M1
A1
A1
AG
N3
N0
(A1)
A1
(M1)
A1A1
N4
(M1)
A1
A1
N0
[17]
8.
(a)
(b)
(c)
Independent (I)
(C2)
Mutually exclusive (M)
(C2)
Neither (N)
(C2)
Note: Award part marks if the candidate shows understanding of I
and/or M
eg I P(A  B) = P(A)P(B)
(M1)
M P(A  B) = P(A) + P(B)
(M1)
[6]
9.
(a)
U(88)
E(32)
H(28)
a
c
b
39
n (E  H) = a + b + c = 88 – 39 = 49
(M1)
n (E  H) = 32 + 28 – b = 49
60 – 49 = b = 11
(A1)
a = 32 – 11 = 21
(A1)
c = 28 – 11 = 17
(A1)
Note: Award (A3) for correct answers with no working.
(b)
(i)
P(E  H) =
11 1

88 8
4
(A1)
21
(ii)
P(HE) =
PH ' E  88

32
PE 
(M1)
88
=
21
(= 0.656)
32
(A1)
OR
Required probability =
(c)
(i)
P(none in economics) =
21
32
(A1)(A1)
56  55  54
88  87  86
3
(M1)(A1)
= 0.253
(A1)
3
 56 
 = 0.258.
 88 
Notes: Award (M0)(A0)(A1)(ft) for 
Award no marks for
(ii)
56  55  54
.
88  88  88
P(at least one) = 1 – 0.253
= 0.747
OR
 32 56 55   32 31 56  32 31 30
    3      
 88 87 86   88 87 86  88 87 86
3
= 0.747
(M1)
(A1)
(M1)
(A1)
5
[12]
10.
P(different colours) = 1 – [P(GG) + P(RR) + P(WW)]
6
5
 10 9 10 9



 
=1–  
 6 25 26 25 26 25 
 210 
=1– 

 650 
44
=
(= 0.677, to 3 sf)
65
(M1)
(A1)
(A1)
(A1) (C4)
OR
P(different colours) = P(GR) + P(RG) + P(GW) + P(WG) + P(RW) + P(WR) (A1)
 10 6   10 10 
= 4    2  
(A1)(A1)
 26 25   26 25 
44
=
(= 0.677, to 3 sf)
(A1) (C4)
65
[4]
11.
p(Red) =
35 7

40 8
p(Black) =
 8  1   7 
p(one black) =     
 1  8   8 
= 0.393 to 3 sf
1
(a)
(i)
(ii)
(b)
5 1

40 8
7
(M1)(A1)
(A1)
p(at least one black) = 1 – p(none)
0
8
 8  1   7 
= 1 –     
 0  8   8 
= 1 – 0.344
= 0.656
400
8
= 50
400 draws: expected number of blacks =
3
(M1)
(A1)
(A1)
3
(M1)
(A1)
2
[8]