Mathematics
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Fluency · Pack A

9.8 Sets, Probability and Games

Answer each question. Show working where needed.

Bronze

  1. Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{3, 5, 7, 9\}$. Write $A \cap B$.

  2. For A = {1,2,3,4,5}, B = {3,5,7,9}, find $A \cup B$.

  3. Let $A$ = set of factors of 12. List $A$.

  4. A fair 6-sided die is rolled. Find P(a number > 4).

  5. A bag has 3 red and 5 blue marbles. P(blue) = ?

  6. A fair coin flipped + die rolled. P(H and 6)?

  7. In a class of 30: 18 study French, 14 Spanish, 8 both. How many study only French?

  8. Two sets: $A = \{1, 2, 3\}$, $B = \{2, 4, 6\}$. State (a) $A \cap B$, (b) $A \cup B$, (c) $A \setminus B$.

  9. Find $P(\text{not red})$ in a bag with 4 red, 6 blue.

  10. A spinner has 4 equal sectors numbered 1-4. Probability of an odd number = ?

Silver

  1. In a survey: n(A) = 24, n(B) = 18, n(A∩B) = 7. Find n(A∪B).

  2. A bag has 3 red, 5 blue. Two drawn without replacement. P(both red)?

  3. A spinner has 4 sectors: £1, £2, £3, £4. Expected winnings?

  4. P(A) = 0.4, P(B) = 0.3, P(A∩B) = 0.1. Find P(A∪B).

  5. A die is rolled 60 times. Expected number of 6s?

  6. A class survey: 18 play football, 15 play basketball, 10 play both. Find n(football ∪ basketball) and n(neither in class of 30).

  7. Two dice are rolled and scores added. P(sum = 7)?

  8. A bag has 5 red, 3 green, 4 blue. P(green)?

  9. In a survey, 60% like tea, 40% like coffee, 20% like both. What % like neither?

  10. A spinner: P(red) = 0.4, P(green) = 0.3, P(blue) = 0.2, P(yellow) = ?

Gold

  1. In a class of 30: 18 play football, 15 play basketball, 10 play both. Given a student plays football, P(also basketball)?

  2. 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.

  3. 3 red, 5 blue marbles. Two drawn without replacement. P(both red)?

  4. A spinner with 8 sectors numbered 1-8 is spun 200 times. Expected primes?

  5. 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**.

  6. Convert non-standard form to ordinary: $4.05 \times 10^5$.

  7. A bag has 4 red and 6 yellow. Two drawn without replacement, both different colours. Find P.

  8. A fair coin is flipped 3 times. P(at least 1 head)?

  9. A game costs £2. P(win £10) = 1/6 (a 6 on a die). Find expected profit.

  10. A bus is on time 70%, 5 min late 25%, 15 min late 5%. Expected delay per journey?

Platinum

  1. 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).

  2. A car-hire company: 60% drive no extra (£0), 30% extra (£6), 10% lots extra (£20). Expected cost?

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

  4. Solve $5^x = 1/125$ for $x$.

  5. 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?

  6. A bus is late 30% of journeys. The passenger takes 100 trips. (a) Expected number of late journeys. (b) Variance and standard deviation.

  7. A lottery: pick 6 numbers from 49. P(all 6 match) = ?

  8. A fair die rolled twice. P(both even) = ?

  9. A bag has 5 red, 3 blue. Drawn with replacement, 3 draws. P(all red)?

  10. A complete deck of 52 cards. Drawn one card. P(king or heart)?