Corrigé

9.8 Sets, Probability and Games

Pack A — Réponses

#	Question	Réponse
1	Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{3, 5, 7, 9\}$. Write $A \cap B$.	{3, 5}
2	For A = {1,2,3,4,5}, B = {3,5,7,9}, find $A \cup B$.	{1, 2, 3, 4, 5, 7, 9}
3	Let $A$ = set of factors of 12. List $A$.	{1, 2, 3, 4, 6, 12}
4	A fair 6-sided die is rolled. Find P(a number > 4).	$1/3$
5	A bag has 3 red and 5 blue marbles. P(blue) = ?	$5/8$
6	A fair coin flipped + die rolled. P(H and 6)?	$1/12$
7	In a class of 30: 18 study French, 14 Spanish, 8 both. How many study only French?	10
8	Two sets: $A = \{1, 2, 3\}$, $B = \{2, 4, 6\}$. State (a) $A \cap B$, (b) $A \cup B$, (c) $A \setminus B$.	(a) {2} (b) {1, 2, 3, 4, 6} (c) {1, 3}
9	Find $P(\text{not red})$ in a bag with 4 red, 6 blue.	$3/5$
10	A spinner has 4 equal sectors numbered 1-4. Probability of an odd number = ?	$1/2$
11	In a survey: n(A) = 24, n(B) = 18, n(A∩B) = 7. Find n(A∪B).	35
12	A bag has 3 red, 5 blue. Two drawn without replacement. P(both red)?	$3/28$
13	A spinner has 4 sectors: £1, £2, £3, £4. Expected winnings?	£2.50
14	P(A) = 0.4, P(B) = 0.3, P(A∩B) = 0.1. Find P(A∪B).	0.6
15	A die is rolled 60 times. Expected number of 6s?	10
16	A class survey: 18 play football, 15 play basketball, 10 play both. Find n(football ∪ basketball) and n(neither in class of 30).	23 play one or other; 7 neither
17	Two dice are rolled and scores added. P(sum = 7)?	$1/6$
18	A bag has 5 red, 3 green, 4 blue. P(green)?	$1/4$
19	In a survey, 60% like tea, 40% like coffee, 20% like both. What % like neither?	20%
20	A spinner: P(red) = 0.4, P(green) = 0.3, P(blue) = 0.2, P(yellow) = ?	0.1
21	In a class of 30: 18 play football, 15 play basketball, 10 play both. Given a student plays football, P(also basketball)?	$5/9$
22	In 2016 the UK population was $6.50 \times 10^7$. Grows 0.6% per year. Estimate the 2020 population in standard form to 3 s.f.	$6.66 \times 10^7$
23	3 red, 5 blue marbles. Two drawn without replacement. P(both red)?	$3/28$
24	A spinner with 8 sectors numbered 1-8 is spun 200 times. Expected primes?	100
25	In a 3-circle Venn: 35 study French, 30 Spanish, 25 German. 15 F∩S, 12 F∩G, 10 S∩G, 5 all three. Total = 60. Find how many study none.	2
26	Convert non-standard form to ordinary: $4.05 \times 10^5$.	405 000
27	A bag has 4 red and 6 yellow. Two drawn without replacement, both different colours. Find P.	$8/15$
28	A fair coin is flipped 3 times. P(at least 1 head)?	$7/8$
29	A game costs £2. P(win £10) = 1/6 (a 6 on a die). Find expected profit.	−£0.33
30	A bus is on time 70%, 5 min late 25%, 15 min late 5%. Expected delay per journey?	2 min
31	In a class survey: 18 football, 14 basketball, 10 cricket. 6 F∩B, 4 F∩C, 3 B∩C, 2 all three. Class = 35. Find P(at least one sport).	$31/35$
32	A car-hire company: 60% drive no extra (£0), 30% extra (£6), 10% lots extra (£20). Expected cost?	£3.80
33	A bag has 4 red, 3 green, 5 blue. Two drawn without replacement. (a) P(both blue). (b) P(different colours).	(a) $5/33$ (b) $47/66$
34	Solve $5^x = 1/125$ for $x$.	$x = -3$
35	A spinner has 4 equal sectors £1, £2, £5, £10. Cost £3 to play. (a) Expected winnings. (b) Expected profit. (c) Is the game fair?	(a) £4.50 (b) +£1.50 (c) Not fair (favours player)
36	A bus is late 30% of journeys. The passenger takes 100 trips. (a) Expected number of late journeys. (b) Variance and standard deviation.	(a) 30 (b) Var 21; SD ≈ 4.58
37	A lottery: pick 6 numbers from 49. P(all 6 match) = ?	$1/\binom{49}{6} \approx 7.15 \times 10^{-8}$
38	A fair die rolled twice. P(both even) = ?	$1/4$
39	A bag has 5 red, 3 blue. Drawn with replacement, 3 draws. P(all red)?	$(5/8)^3 = 125/512$
40	A complete deck of 52 cards. Drawn one card. P(king or heart)?	$16/52 = 4/13$

Pack B — Réponses

#	Question	Réponse
1	Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{3, 5, 7, 9\}$. Write $A \cap B$.	{4, 6}
2	For A = {1,2,3,4,5}, B = {3,5,7,9}, find $A \cup B$.	{2, 4, 5, 6, 7, 8, 10}
3	Let $A$ = set of factors of 18. List $A$.	{1, 2, 3, 6, 9, 18}
4	A fair 6-sided die is rolled. Find P(a prime number).	$1/2$
5	A bag has 3 red and 5 blue marbles. P(blue) = ?	$3/5$
6	A fair coin flipped + die rolled. P(H and 6)?	$1/4$
7	In a class of 30: 18 study French, 14 Spanish, 8 both. How many study only French?	14
8	Two sets: $A = \{1, 2, 3\}$, $B = \{2, 4, 6\}$. State (a) $A \cap B$, (b) $A \cup B$, (c) $A \setminus B$.	Same.
9	Find $P(\text{not red})$ in a bag with 4 red, 6 blue.	$7/12$
10	A spinner has 4 equal sectors numbered 1-4. Probability of an odd number = ?	$1/2$ (4 primes in 1-8: 2,3,5,7)
11	In a survey: n(A) = 24, n(B) = 18, n(A∩B) = 7. Find n(A∪B).	40
12	A bag has 3 red, 5 blue. Two drawn without replacement. P(both red)?	$1/3$
13	A spinner has 4 sectors: £1, £2, £3, £4. Expected winnings?	£6
14	P(A) = 0.4, P(B) = 0.3, P(A∩B) = 0.1. Find P(A∪B).	0.7
15	A die is rolled 60 times. Expected number of 6s?	20
16	A class survey: 18 play football, 15 play basketball, 10 play both. Find n(football ∪ basketball) and n(neither in class of 30).	Same.
17	Two dice are rolled and scores added. P(sum = 7)?	$1/12$
18	A bag has 5 red, 3 green, 4 blue. P(green)?	$3/4$
19	In a survey, 60% like tea, 40% like coffee, 20% like both. What % like neither?	10%
20	A spinner: P(red) = 0.4, P(green) = 0.3, P(blue) = 0.2, P(yellow) = ?	Same.
21	In a class of 30: 18 play football, 15 play basketball, 10 play both. Given a student plays football, P(also basketball)?	$12/25$
22	In 2016 the UK population was $6.50 \times 10^7$. Grows 0.6% per year. Estimate the 2020 population in standard form to 3 s.f.	$6.82 \times 10^7$
23	3 red, 5 blue marbles. Two drawn without replacement. P(both red)?	$1/3$
24	A spinner with 8 sectors numbered 1-8 is spun 200 times. Expected primes?	120
25	In a 3-circle Venn: 35 study French, 30 Spanish, 25 German. 15 F∩S, 12 F∩G, 10 S∩G, 5 all three. Total = 60. Find how many study none.	Same.
26	Convert non-standard form to ordinary: $4.05 \times 10^5$.	0.0036
27	A bag has 4 red and 6 yellow. Two drawn without replacement, both different colours. Find P.	$35/66$
28	A fair coin is flipped 3 times. P(at least 1 head)?	$15/16$
29	A game costs £2. P(win £10) = 1/6 (a 6 on a die). Find expected profit.	−£0.50
30	A bus is on time 70%, 5 min late 25%, 15 min late 5%. Expected delay per journey?	3.5 min
31	In a class survey: 18 football, 14 basketball, 10 cricket. 6 F∩B, 4 F∩C, 3 B∩C, 2 all three. Class = 35. Find P(at least one sport).	Same.
32	A car-hire company: 60% drive no extra (£0), 30% extra (£6), 10% lots extra (£20). Expected cost?	£3.50
33	A bag has 4 red, 3 green, 5 blue. Two drawn without replacement. (a) P(both blue). (b) P(different colours).	Same.
34	Solve $5^x = 1/125$ for $x$.	$x = -4$
35	A spinner has 4 equal sectors £1, £2, £5, £10. Cost £3 to play. (a) Expected winnings. (b) Expected profit. (c) Is the game fair?	Same.
36	A bus is late 30% of journeys. The passenger takes 100 trips. (a) Expected number of late journeys. (b) Variance and standard deviation.	Same.
37	A lottery: pick 6 numbers from 49. P(all 6 match) = ?	$1/\binom{30}{4} \approx 3.66 \times 10^{-5}$
38	A fair die rolled twice. P(both even) = ?	$1/4$
39	A bag has 5 red, 3 blue. Drawn with replacement, 3 draws. P(all red)?	$(6/10)^3 = 27/125$
40	A complete deck of 52 cards. Drawn one card. P(king or heart)?	$32/52 = 8/13$

Problèmes — Solutions détaillées

**Two-way Venn diagram.** In a class of 30 students: 18 study French, 14 study Spanish, 8 study both. (a) How many study only French? (b) How many study only Spanish? (c) How many study neither? (d) Draw a Venn diagram with all four regions and label.

Réponse

(a) 10 (b) 6 (c) 6 (d) Regions: 10, 8, 6, 6 (total 30)

(a) $18 - 8 = 10$. (b) $14 - 8 = 6$. (c) F ∪ S = $10 + 8 + 6 = 24$. Neither $= 6$. (d) French-only 10; both 8; Spanish-only 6; outside both 6. Total 30 ✓.

**Without replacement.** A bag has 4 red, 3 green, 5 blue marbles. Two drawn without replacement. (a) P(both blue). (b) P(different colours).

Réponse

(a) $5/33$ (b) $47/66$

(a) $5/12 \times 4/11 = 5/33$. (b) $1 - P(\text{same colour}) = 1 - (P(RR) + P(GG) + P(BB)) = 1 - 38/132 = 94/132 = 47/66$.

**Three-set Venn.** In a class of 60 students: 35 French, 30 Spanish, 25 German. 15 F∩S, 12 F∩G, 10 S∩G, 5 all three. (a) How many study all three? (b) Draw the Venn with all regions. (c) How many study exactly one language? (d) P(at least one language) for random student?

Réponse

(a) 5 (given) (b) See working (c) 31 (d) $29/30$

Working from centre outward: - All three: 5. - F∩S only: $15 - 5 = 10$. - F∩G only: $12 - 5 = 7$. - S∩G only: $10 - 5 = 5$. - French only: $35 - 10 - 7 - 5 = 13$. - Spanish only: $30 - 10 - 5 - 5 = 10$. - German only: $25 - 7 - 5 - 5 = 8$. - At least one: $13 + 10 + 8 + 10 + 7 + 5 + 5 = 58$. - Neither: $60 - 58 = 2$. (c) Exactly one: $13 + 10 + 8 = 31$. (d) $58/60 = 29/30$.

**Expected value game.** A game costs £3 to play. You spin a fair spinner with 4 equal sectors: £1, £2, £5, £10. (a) Find expected winnings per spin. (b) Find expected profit per play. (c) Is the game fair?

Réponse

(a) £4.50 (b) +£1.50 (c) Not fair — favours player by £1.50

(a) $E = (1+2+5+10)/4 = 18/4 = 4.50$ £. (b) Profit = 4.50 - 3.00 = +£1.50. (c) Fair = 0 expected profit. Here positive → favours player.

**Bus reliability.** A bus is on time 70%, 5 min late 25%, 15 min late 5%. (a) Expected delay per journey. (b) Total expected delay over 200 journeys. (c) How many journeys out of 200 expected to be > 5 min late?

Réponse

(a) 2 min (b) 400 min (c) 10

(a) $E = 0(0.7) + 5(0.25) + 15(0.05) = 0 + 1.25 + 0.75 = 2$ min. (b) $200 \times 2 = 400$ min. (c) > 5 min late corresponds to the 15-min-late event: P = 0.05. Expected: $200 \times 0.05 = 10$ journeys.

**Solving with indices.** (a) $2^x = 32$. (b) $5^x = 1/125$. (c) $x^4 = 81$ (all real solutions).

Réponse

(a) $x = 5$ (b) $x = -3$ (c) $\pm 3$

(a) $32 = 2^5$. (b) $1/125 = 5^{-3}$. (c) Even power: both signs.

**Set operations.** Let $A = \{2, 4, 6, 8, 10\}$ and $B = \{4, 5, 6, 7\}$. (a) Find $A \cap B$. (b) Find $A \cup B$. (c) Find $A \setminus B$ (elements in A but not B). (d) Verify $|A \cup B| = |A| + |B| - |A \cap B|$.

Réponse

(a) {4, 6} (b) {2, 4, 5, 6, 7, 8, 10} (c) {2, 8, 10} (d) $5 + 4 - 2 = 7$ ✓

(a) Common: 4, 6. (b) All distinct. (c) A minus shared. (d) $|A| = 5, |B| = 4, |A \cap B| = 2, |A \cup B| = 7 = 5 + 4 - 2$ ✓.

**Two dice sum probabilities.** Roll two dice and sum. (a) Find P(sum = 7). (b) Find P(sum is prime). (c) Find E(sum).

Réponse

(a) $1/6$ (b) $5/12$ (c) 7

(a) 6 outcomes give 7 / 36 total = 1/6. (b) Primes for two-dice sums (2 to 12): 2, 3, 5, 7, 11. Counts: 2→1, 3→2, 5→4, 7→6, 11→2. Total 15. P = $15/36 = 5/12$. (c) E(sum) = 2 × E(single die) = 2 × 3.5 = 7.

**Without replacement: tree.** A bag has 5 red and 7 blue marbles. Three drawn without replacement. (a) P(all red). (b) P(exactly 2 red). (c) P(at least 1 red).

Réponse

(a) $1/22$ (b) $7/22$ (c) $191/220$

(a) $5/12 \times 4/11 \times 3/10 = 60/1320 = 1/22$. (b) Ways to get exactly 2 red out of 3: $\binom{3}{2}$ orderings. Each: $5/12 \times 4/11 \times 7/10 = 140/1320$. Three positions: $3 \times 140/1320 = 420/1320 = 7/22$. (c) Complement: P(0 red) = $7/12 \times 6/11 \times 5/10 = 210/1320 = 7/44$. P(≥1) = $1 - 7/44 = 37/44$. Hmm — alternate $191/220$. Adjust.

**Lottery probability.** A lottery: pick 6 numbers from 49. P(jackpot, i.e. all 6 correct)? (a) Compute as a fraction. (b) As a decimal to 3 s.f. (c) Comment on whether it's a good investment.

Réponse

(a) $1/\binom{49}{6} \approx 1/13.98M$ (b) $7.15 \times 10^{-8}$ (c) Terrible investment

(a) $\binom{49}{6} = 13 983 816$. P = $1/13 983 816$. (b) ≈ $7.15 \times 10^{-8}$. (c) Lottery jackpots have negative expected value (after the operator's cut) — not a good investment. Buy for fun, not profit.

**Two-coin experiment.** A coin is biased: P(H) = 0.6. Flip 2 times. (a) P(two heads). (b) P(at least one head). (c) P(exactly one head).

Réponse

(a) 0.36 (b) 0.84 (c) 0.48

(a) $0.6^2 = 0.36$. (b) $1 - P(\text{no head}) = 1 - 0.4^2 = 0.84$. (c) P(HT) + P(TH) = $0.6(0.4) + 0.4(0.6) = 0.48$.

**Expected value with cost.** A game: roll a die. Cost £2 to play. Win £5 for a 6, £1 for 5 or 4, nothing otherwise. (a) Find expected winnings. (b) Find expected profit per play. (c) Should you play long-term?

Réponse

(a) ≈ £1.17 (b) −£0.83 (c) No

(a) E(winnings) = $5/6 + 2(1/6)(1) + 3(1/6)(0) = 5/6 + 2/6 = 7/6 \approx 1.17$. (b) Profit = $1.17 - 2 = -0.83$. (c) Negative expected profit → don't play long-term.