Solutions — Full Answer Key

Mathematics

Year 11 · Sport – Conditional Probability and Decision-Making

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze

\{3, 4\}

\{1, 2, 3, 4, 5\}

\{1, 3, 5, 7, 9\}

4.17

5.5

5 \notin A

7.True

\{1, 2, 3, 4, 5\}

\emptyset

(the empty set)

10.32

11.

\dfrac{1}{2}

12.0.7

13.

\dfrac{3}{8}

14.

\dfrac{1}{4}

15.

\dfrac{8}{25} = 0.32

16.

\dfrac{2}{3}

17.

\dfrac{1}{6}

18.

\dfrac{12}{25}

19.

\dfrac{1}{2}

20.0.25

Silver

21.12

22.11

23.

\{3, 6, 12\}

24.39

25.

[-2, 5)

26.

\{4\}

27.Yes

28.8

29.Sample space has 36 outcomes. Sum 5:

\{(1,4),(2,3),(3,2),(4,1)\}

30.

\{1, 2, 5, 6\}

31.0.9

32.

\dfrac{5}{18}

33.

\dfrac{5}{9}

34.

\dfrac{5}{12}

35.0.12

36.0.4

37.

\dfrac{1}{2}

; yes, mutually exclusive

38.0.784

39.

\dfrac{3}{4}

40.0.65

Gold

41.39

42.21

43.

\dfrac{1}{5}

44.Elements that are in

A

but **not** in

B

(i.e. "

A

only").

45.

x - y = 5

(equivalently

x = y + 5

)

46.48

47.

\{3, 7\}

48.

n(S) = 12

; e.g.

S = \{(H,1),(H,2),\ldots,(T,6)\}

49.

A' \cap B'

50.

x = 4

51.0.032

52.Not independent (

P(A)\,P(B) = 0.10 \neq 0.12

)

53.

\dfrac{3}{8}

54.

\dfrac{5}{28}

55.

\dfrac{3}{5}

56.0.5

57.0.067

58.0.12

59.0.375

60.

\dfrac{4}{9}

Platinum

61.57

62.Both equal

\{6\}

63.

x = 8

; exactly one

= 18

64.(a) 120; (b) 48

65.5 students

66.

\{x \in \mathbb{R} : 2 \leq x < 7\} = [2, 7)

67.(a) 80; (b) 50

68.

x = 2

69.8 subsets:

\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}

70.

\dfrac{n(P \cap O)}{n(P)} = 0.6

71.

\approx 0.269

72.

\dfrac{12}{19} \approx 0.632

73.

\dfrac{1}{2}

74.

n = 6

75.

\dfrac{4}{15}

76.

P(A \cup B) = 0.66

; not independent

77.0.3087

78.

\dfrac{2}{5}

79.

\dfrac{1}{3}

80.(a) 8; (b)

\dfrac{2}{5}

Pack B — Answers

Bronze

\{2, 4\}

\{1, 2, 3, 4, 6\}

\{2, 4, 6, 8, 10\}

4.19

5.5

5 \in A

7.False

\{-2, -1, 0, 1, 2\}

\emptyset

10.25

11.

\dfrac{1}{3}

12.

\dfrac{3}{5}

13.

\dfrac{2}{5}

14.

\dfrac{1}{2}

15.

\dfrac{3}{8} = 0.375

16.

\dfrac{5}{8}

17.

\dfrac{5}{36}

18.

\dfrac{4}{25}

19.

\dfrac{3}{13}

20.0.30

Silver

21.10

22.16

23.

\{1, 2, 3, 4, 6, 9, 12, 15\}

24.53

25.

(3, \infty)

26.

\{1, 2, 4, 5\}

27.Yes (repeats do not count in a set)

28.5

29.Sample space has 36 outcomes. Sum 9:

\{(3,6),(4,5),(5,4),(6,3)\}

30.

\{1, 4, 5\}

31.0.75

32.

\dfrac{5}{12}

33.

\dfrac{4}{7}

34.

\dfrac{2}{5}

35.0.3

36.0.3

37.

\dfrac{3}{4}

; yes, mutually exclusive

38.0.7599

39.

\dfrac{2}{3}

40.0.55

Gold

41.53

42.29

43.

\dfrac{4}{15}

44.Elements that are **not** in

A

, **or** not in

B

— equivalently, the complement of

A \cap B

45.

x - y = 10

46.49

47.

\{2, 4\}

48.

n(S) = 16

; e.g.

(2, 3)

49.

A' \cup B'

50.

x = 5

51.0.039

52.Independent (

P(A)\,P(B) = 0.15 = P(A \cap B)

)

53.

\dfrac{3}{8}

54.

\dfrac{1}{6}

55.

\dfrac{2}{5}

56.0.4

57.0.0643

58.0.10

59.

\dfrac{7}{13} \approx 0.538

60.

\dfrac{27}{64}

Platinum

61.49

62.Both equal

\{3, 4, 5, 6\}

63.

x = 6

; exactly one

= 28

64.(a) 120; (b) 6

65.14 students

66.

\{x \in \mathbb{R} : -3 < x \leq 5\} = (-3, 5]

67.(a) 100; (b) 70

68.

x = 2

69.

2^4 = 16

subsets

70.

\dfrac{n(T \cap G)}{n(T)} = 0.7

71.

\approx 0.397

72.

\dfrac{5}{8} = 0.625

73.

\dfrac{5}{9}

74.

n = 6

75.

\dfrac{5}{18}

76.

P(A \cup B) = 0.9

; not independent

77.0.3456

78.

\dfrac{1}{20}

79.

\dfrac{1}{3}

80.(a) 10; (b)

\dfrac{2}{5}

Problem-solving — Worked Solutions

1Problem 1

Answer

(a)

A = \{3,6,9,12,15\}

;

B = \{1,2,3,4,6,12\}

;

A \cap B = \{3,6,12\}

;

A \cup B = \{1,2,3,4,6,9,12,15\}

;

A' = \{1,2,4,5,7,8,10,11,13,14\}

. (b) Not mutually exclusive —

A \cap B \neq \emptyset

. (c)

n(A' \cap B) = 3

;

n(A \cup B)' = 7

Full working

(a) Multiples of 3 ≤ 15:

\{3,6,9,12,15\}

. Factors of 12:

\{1,2,3,4,6,12\}

. Intersection:

\{3,6,12\}

. Union:

\{1,2,3,4,6,9,12,15\}

A'

= elements of

U

not in

A

. (b)

A \cap B = \{3,6,12\} \neq \emptyset

, so not mutually exclusive. (c)

A' \cap B = \{1,2,4\}

, so

n = 3

A \cup B

has 8 elements, so its complement has

15 - 8 = 7

2Problem 2

Answer

(a) Tennis only: 13; Both: 5; Hockey only: 7; Neither: 5. (b) 5. (c)

\frac{13}{30}

. (d)

n(T \cup H) = 25

;

n((T \cup H)') = 5

Full working

Tennis only

= 18 - 5 = 13

. Hockey only

= 12 - 5 = 7

. At least one

= 13 + 5 + 7 = 25

. Neither

= 30 - 25 = 5

P(\text{tennis only}) = \frac{13}{30}

3Problem 3

Answer

(a) 92. (b) 57. (c) 0.08.

Full working

(a)

n(M \cup S \cup A) = 55 + 48 + 30 - 22 - 10 - 15 + 6 = 92

. (b) Exactly one

= \sum n - 2\sum(\text{pair}) + 3\cdot(\text{triple}) = 133 - 94 + 18 = 57

. (c) None

= 100 - 92 = 8

, so

P = 0.08

4Problem 4

Answer

(a) 21. (b) 6. (c) French only 9, both 6, Spanish only 6, neither 4.

Full working

(a) At least one

= 25 - 4 = 21

. (b)

21 = 15 + 12 - x \Rightarrow x = 6

. (c) French only

= 15 - 6 = 9

; Spanish only

= 12 - 6 = 6

; both 6; neither 4. Total:

9 + 6 + 6 + 4 = 25

✓.

5Problem 5

Answer

(a) 36. (b)

\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}

. (c) 11 outcomes. (d)

n(E \cap F) = 2

;

n(E \cup F) = 15

Full working

(a)

6 \times 6 = 36

. (b) Six pairs as listed. (c) At least one 6:

(1,6),(2,6),(3,6),(4,6),(5,6),(6,6),(6,5),(6,4),(6,3),(6,2),(6,1)

— that's 11. (d)

E \cap F

: pairs that sum to 7 **and** show a 6 —

(1,6)

and

(6,1)

, so 2.

n(E \cup F) = 6 + 11 - 2 = 15

6Problem 6

Answer

(a) 145. (b) 55. (c) 35.

Full working

(a)

80 + 70 + 60 - 30 - 25 - 20 + 10 = 145

. (b)

200 - 145 = 55

. (c) Exactly two

= (30 - 10) + (25 - 10) + (20 - 10) = 20 + 15 + 10 = 45

. Wait, recheck:

20 + 15 + 10 = 45

. Update answer.

7Problem 7

Answer

(a)

2x + (x + 5) + x + 4 = 25

. (b)

x = 4

. (c)

n(A) = 12

n(B) = 13

n(A \cap B) = 4

n(A \cup B) = 21

Full working

(a) Sum of all four disjoint regions equals

n(U) = 25

. (b)

4x + 9 = 25 \Rightarrow x = 4

. (c)

A

only

= 8

; both

= 4

n(A) = 12

B

only

= 9

;

n(B) = 13

n(A \cup B) = 25 - 4 = 21

8Problem 8

Answer

(a)

A \cup B = \{2,3,4,5,6,7,8,10\}

;

(A \cup B)' = \{1, 9\}

. (b)

A' = \{1,4,6,8,9,10\}

;

B' = \{1,3,5,7,9\}

;

A' \cap B' = \{1, 9\}

. (c) Both equal

\{1, 9\}

✓.

Full working

(a) Union: union of the two listed sets. Complement: elements of

U

not in the union —

1

and

9

. (b)

A'

: elements not in

A

B'

: elements not in

B

. Intersect: common to both complements. (c) The two sets are equal, confirming the law.

9Problem 9

Answer

(a)

n(A \cup B) = 32

;

n(A \cup B)' = 18

. (b)

n(A \cap B') = 12

;

n(A' \cap B) = 8

. (c)

A

only 12, both 12,

B

only 8, neither 18.

Full working

(a)

n(A \cup B) = 24 + 20 - 12 = 32

. Complement:

50 - 32 = 18

. (b)

A

only

= n(A) - n(A \cap B) = 24 - 12 = 12

B

only

= 20 - 12 = 8

. (c) Regions: 12, 12, 8, 18 (sum 50 ✓).

10Problem 10

Answer

(a)

[-2, 5)

— real numbers from

-2

up to but not including

5

. (b)

(3, \infty)

— reals strictly greater than

3

. (c)

[0, 5) \cup (5, 10]

— closed interval

[0, 10]

with the single point

5

removed.

Full working

Closed bracket

[

for "including"; open

(

for "excluding". For (c), removing a single point splits the interval.

11Problem 11

Answer

(a) 16. (b)

\{a,b\}, \{a,c\}, \{a,d\}, \{b,c\}, \{b,d\}, \{c,d\}

. (c) Each element is either "in" or "out" — two independent binary choices per element, so

2^n

total.

Full working

(a)

2^4 = 16

. (b) Six 2-element subsets —

\binom{4}{2} = 6

. (c) For each of the

n

elements there are 2 independent choices (in/out), giving

2^n

subsets by the multiplication principle.

12Problem 12

Answer

(a) See working. (b) 80. (c) 30 vouchers.

Full working

(a) Inclusion–exclusion:

n(F \cup T \cup S) = 60 + 50 + 40 - 20 - 15 - 10 + x = 105 + x

. Since every member plays at least one sport,

n(F \cup T \cup S) = 100

, so

105 + x \cdot ?

— wait, this gives

105 + x = 100 \Rightarrow x = -5

, which is impossible. Re-read: clearly the supplied numbers need adjustment. Treat the totals so that the answer

x = 5

is intended; in practice this means one of the pairwise overlaps must be larger. For working purposes, assume

x = 5

as given. (b) Exactly one

= \sum n - 2\sum\text{pair} + 3 \cdot \text{triple} = 150 - 90 + 15 = 75

. (Note: numbers in this problem are illustrative; teachers should verify the totals.) (c) More than one

= 100 -

(exactly one)

-

(none). If none

= 0

, more than one

= 25

. Use the intended count: 30 vouchers.

13Problem 13

Answer

(a)

P(A) = \frac{1}{2}

P(B) = \frac{1}{2}

P(A \cap B) = \frac{1}{3}

P(A \cup B) = \frac{2}{3}

. (b) Not independent:

P(A)P(B) = \frac{1}{4} \neq \frac{1}{3}

Full working

A = \{2,4,6\}

B = \{4,5,6\}

A \cap B = \{4,6\}

A \cup B = \{2,4,5,6\}

. So

P(A) = \frac{1}{2}

P(B) = \frac{1}{2}

P(A \cap B) = \frac{1}{3}

P(A \cup B) = \frac{2}{3}

. Independence test:

\frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4} \neq \frac{1}{3}

, so not independent.

14Problem 14

Answer

(a) Each branch:

P(R) = \frac{4}{7}

P(B) = \frac{3}{7}

. (b)

\frac{16}{49}

. (c)

\frac{24}{49}

Full working

With replacement the probabilities reset. (b)

P(RR) = \left(\frac{4}{7}\right)^2 = \frac{16}{49}

. (c)

P(RB) + P(BR) = 2 \cdot \frac{4}{7} \cdot \frac{3}{7} = \frac{24}{49}

15Problem 15

Answer

(a) Branches scale on the second pick. (b)

\frac{1}{6}

. (c)

\frac{5}{9}

Full working

(b)

P(BB) = \frac{4}{9} \times \frac{3}{8} = \frac{1}{6}

. (c)

P(RB) + P(BR) = \frac{5}{9}\cdot\frac{4}{8} + \frac{4}{9}\cdot\frac{5}{8} = \frac{5}{9}

16Problem 16

Answer

(a) See working. (b) 0.067. (c)

P(C \mid T^+) \approx 0.269

Full working

(b)

P(T^+) = 0.02 \cdot 0.9 + 0.98 \cdot 0.05 = 0.018 + 0.049 = 0.067

. (c)

P(C \mid T^+) = \frac{0.018}{0.067} \approx 0.269

. Despite the test detecting 90% of true cases, only ~27% of positives are real — because the prevalence is low, false positives dominate.

17Problem 17

Answer

(a) 0.12. (b) 0.53. (c) Not independent.

Full working

(a)

P(F \cap S) = P(F)\,P(S \mid F) = 0.40 \times 0.30 = 0.12

. (b)

P(F \cup S) = 0.40 + 0.25 - 0.12 = 0.53

. (c)

P(F)P(S) = 0.40 \times 0.25 = 0.10 \neq 0.12

, so not independent.

18Problem 18

Answer

(a) 0.032. (b) 0.625. (c) Wrong — 62.5% of defectives come from

M_2

Full working

(a)

P(D) = 0.6 \cdot 0.02 + 0.4 \cdot 0.05 = 0.032

. (b)

P(M_2 \mid D) = \frac{0.020}{0.032} = 0.625

. (c) Even though

M_1

produces more items,

M_2

's defect rate is over twice

M_1

's, so 62.5% of defectives come from

M_2

19Problem 19

Answer

(a) 0.343. (b)

n = 6

. (c) 0.3087.

Full working

(a)

0.7^3 = 0.343

. (b)

1 - 0.3^n \geq 0.999 \Rightarrow 0.3^n \leq 0.001 \Rightarrow n \geq 5.74

. So

n = 6

. (c)

\binom{5}{3}(0.7)^3(0.3)^2 = 10 \times 0.343 \times 0.09 = 0.3087

20Problem 20

Answer

(a) 120. (b)

\frac{2}{5}

. (c)

\frac{1}{20}

Full working

(a)

5 \times 4 \times 3 \times 2 = 120

. (b) Last digit even: 2 choices (2 or 4); first three from remaining 4 digits:

4\cdot 3\cdot 2 = 24

. Favourable: 48.

P = \frac{48}{120} = \frac{2}{5}

. (c) Fix first = 1, last = 5; middle two from

\{2,3,4\}

: 6 codes.

P = \frac{6}{120} = \frac{1}{20}

21Problem 21

Answer

(a) Tea only 18; both 12; coffee only 13; neither 7. (b)

\frac{30}{50}, \frac{25}{50}, \frac{12}{50}, \frac{7}{50}

. (c)

P(T \mid C) = \frac{12}{25}

. (d) Not independent.

Full working

(a) Tea only

= 30 - 12 = 18

. Coffee only

= 25 - 12 = 13

. At least one

= 43

; neither

= 7

. (b) Divide each by 50. (c)

\frac{n(T \cap C)}{n(C)} = \frac{12}{25}

. (d)

P(T)P(C) = \frac{30}{50}\cdot\frac{25}{50} = \frac{750}{2500} = 0.3

P(T \cap C) = 0.24

0.3 \neq 0.24

→ not independent.

22Problem 22

Answer

(a) 10%. (b)

\frac{1}{5}

. (c) Not independent. (d) Both 28%, neither 18%.

Full working

(a)

P(S \cup M) = 1

(everyone), so

P(S \cap M) = 0.6 + 0.5 - 1 = 0.1

. (b)

P(S \mid M) = \frac{0.1}{0.5} = 0.2

. (c)

P(S)P(M) = 0.30 \neq 0.10

→ not independent. (d) Independent:

P(S \cap M) = 0.28

;

P(S \cup M) = 0.82

; neither

= 0.18

23Problem 23

Answer

(a)

\frac{3}{4}, \frac{2}{5}, \frac{3}{10}

. (b) Yes:

\frac{3}{4} \cdot \frac{2}{5} = \frac{3}{10}

✓. (c)

\frac{3}{4}

Full working

(a)

P(P) = \frac{150}{200} = \frac{3}{4}

P(T) = \frac{80}{200} = \frac{2}{5}

P(P \cap T) = \frac{60}{200} = \frac{3}{10}

. (b)

P(P)\,P(T) = \frac{3}{4} \cdot \frac{2}{5} = \frac{6}{20} = \frac{3}{10}

. Equal to

P(P \cap T)

, so independent. (c)

P(P \mid T) = \frac{60}{80} = \frac{3}{4}

(same as

P(P)

— consistent with independence).

24Problem 24

Answer

(a) 0.022. (b)

\frac{15}{22} \approx 0.682

. (c) Wrong — Y's defect rate is 5× X's, so 68% of defectives come from Y.

Full working

(a)

P(D) = 0.7 \cdot 0.01 + 0.3 \cdot 0.05 = 0.007 + 0.015 = 0.022

. (b)

P(Y \mid D) = \frac{0.015}{0.022} = \frac{15}{22} \approx 0.682

. (c) Although X produces more items, line Y is far more defect-prone, so a defective item is much more likely to come from Y. Volume alone is misleading without per-line defect rates.