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Problem-solving Pack
MathematicsYear 9 · 9.8 Sets, Probability and Games
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
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.
(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.
Working space
2Problem 2 of 12
Without replacement. A bag has 4 red, 3 green, 5 blue marbles. Two drawn without replacement.
(a) P(both blue).
(b) P(different colours).
(a) P(both blue).
(b) P(different colours).
Working space
3Problem 3 of 12
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?
(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?
Working space
4Problem 4 of 12
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?
(a) Find expected winnings per spin.
(b) Find expected profit per play.
(c) Is the game fair?
Working space
5Problem 5 of 12
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?
(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?
Working space
6Problem 6 of 12
Solving with indices.
(a) .
(b) .
(c) (all real solutions).
(a) .
(b) .
(c) (all real solutions).
Working space
7Problem 7 of 12
Set operations. Let and .
(a) Find .
(b) Find .
(c) Find (elements in A but not B).
(d) Verify .
(a) Find .
(b) Find .
(c) Find (elements in A but not B).
(d) Verify .
Working space
8Problem 8 of 12
Two dice sum probabilities. Roll two dice and sum.
(a) Find P(sum = 7).
(b) Find P(sum is prime).
(c) Find E(sum).
(a) Find P(sum = 7).
(b) Find P(sum is prime).
(c) Find E(sum).
Working space
9Problem 9 of 12
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).
(a) P(all red).
(b) P(exactly 2 red).
(c) P(at least 1 red).
Working space
10Problem 10 of 12
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.
(a) Compute as a fraction.
(b) As a decimal to 3 s.f.
(c) Comment on whether it's a good investment.
Working space
11Problem 11 of 12
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).
(a) P(two heads).
(b) P(at least one head).
(c) P(exactly one head).
Working space
12Problem 12 of 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?
(a) Find expected winnings.
(b) Find expected profit per play.
(c) Should you play long-term?
Working space