Problems — 9.8 Sets, Probability and Games

1

**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.

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2

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

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3

**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?

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4

**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?

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5

**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?

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6

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

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7

**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 $|A \cup B| = |A| + |B| - |A \cap B|$.

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8

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

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9

**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).

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10

**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.

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11

**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).

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12

**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?

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