Mathematics • Year 8 • Unit 4 • Lesson 20

Probability in the Real World — Full Review

Apply every Unit 4 tool to real-world contexts: airline lateness data, sport club membership Venn, lucky-dip with replacement, two-spinner game design, and weather-forecast simulation.

Apply · Real-World Maths

1. Word problems

Each problem uses ideas from across Unit 4. Choose the right tool: sample space, Venn, tree, experimental, or complement. Show working.

1.1 — Airline lateness. An airline records 250 flights and finds 35 of them were late.

(a) Find the experimental P(flight is late).
(b) If the airline runs 1,200 flights next month, how many late flights are expected?
(c) Use the complement rule to find P(flight is on time). 4 marks

Stuck? P(late) = 35/250. Then expected = P × 1200 and P(on time) = 1 − P(late).

1.2 — Sport club membership. Of 80 Year 8 students: 35 are in the soccer club (S), 25 are in the athletics club (A), 12 are in both, and the rest are in neither.

(a) Draw a Venn diagram with all four regions filled.
(b) Find P(in at least one club).
(c) Find P(in athletics only). 4 marks

Stuck on (a)? Start with the intersection (12), then S only = 35 − 12, A only = 25 − 12.

1.3 — School fair lucky dip. A lucky-dip bag has 6 small prizes and 14 dud tickets (20 total). Two draws are made with replacement.

(a) Draw a tree and find P(both wins).
(b) Find P(at least one win).
(c) Verify all endpoints sum to 1. 4 marks

Stuck on (b)? Use the complement: P(at least one win) = 1 − P(both lose).

1.4 — Two-spinner game. A school carnival uses two spinners. Spinner 1 has 3 equal sectors (R, G, B). Spinner 2 has 4 equal sectors (1, 2, 3, 4). A win is "Red AND an even number".

(a) State |S| using the counting principle.
(b) List all favourable outcomes.
(c) Find P(winning). If 240 students play and each pays $1, with a $3 prize per win, how much net profit does the carnival expect? 4 marks

Stuck on (c)? Expected wins = P(win) × 240. Expected payout = expected wins × $3. Net profit = $240 − payout.

1.5 — Rain forecast simulation. A weather model gives P(rain on Saturday) = 0.4 and P(rain on Sunday) = 0.5 (treat as independent).

(a) Draw a tree (R/N for each day).
(b) Find P(rain both days), P(rain only Saturday), P(rain only Sunday), P(no rain either day).
(c) Find P(at least one rainy day) using the complement. 4 marks

Stuck on (c)? P(at least one rainy) = 1 − P(no rain either day).

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate is stuck and asks: "I have a probability problem — should I use a Venn diagram or a tree diagram?" In your own words explain (i) when a Venn diagram is the right tool (e.g., kind of data), (ii) when a tree diagram is the right tool (e.g., kind of experiment), (iii) give one specific everyday example of each. Use the term compound event somewhere in your answer.

Stuck? Revisit lesson § "Spot the Trap" — Venn for overlapping groups/sets, tree for sequential stages of a compound event.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Airline lateness

(a) P(late) ≈ 35/250 = 7/50 = 0.14.
(b) Expected = 0.14 × 1200 = 168 late flights.
(c) P(on time) = 1 − 0.14 = 0.86.

1.2 — Sport club Venn

(a) S only = 35 − 12 = 23; Both = 12; A only = 25 − 12 = 13; Neither = 80 − 23 − 12 − 13 = 32. Check: 23 + 12 + 13 + 32 = 80 ✓.
(b) P(in at least one) = (23 + 12 + 13) / 80 = 48/80 = 3/5.
(c) P(athletics only) = 13/80.

1.3 — Lucky dip with replacement

P(W) = 6/20 = 3/10; P(L) = 14/20 = 7/10.
(a) P(both W) = 3/10 × 3/10 = 9/100.
(b) P(both L) = 7/10 × 7/10 = 49/100. P(at least one win) = 1 − 49/100 = 51/100.
(c) Endpoints: WW = 9/100; WL = 21/100; LW = 21/100; LL = 49/100. Sum = (9 + 21 + 21 + 49)/100 = 100/100 = 1 ✓.

1.4 — Two-spinner game

(a) |S| = 3 × 4 = 12.
(b) Favourable (R, even): (R, 2), (R, 4) → 2 outcomes.
(c) P(win) = 2/12 = 1/6. Expected wins in 240 plays = (1/6) × 240 = 40. Expected payout = 40 × $3 = $120. Net profit = $240 − $120 = $120.

1.5 — Rain forecast tree

(a) Saturday: P(R) = 0.4, P(N) = 0.6. Sunday: P(R) = 0.5, P(N) = 0.5.
(b) P(both rain) = 0.4 × 0.5 = 0.20. P(only Sat) = 0.4 × 0.5 = 0.20. P(only Sun) = 0.6 × 0.5 = 0.30. P(no rain either) = 0.6 × 0.5 = 0.30. Sum = 0.20 + 0.20 + 0.30 + 0.30 = 1.00 ✓.
(c) P(at least one rainy day) = 1 − 0.30 = 0.70.

2.1 — Explain your thinking (sample response)

Use a Venn diagram when you have overlapping groups or sets within a single population — for example, surveying 100 students about who plays sport AND who plays an instrument, where some belong to both groups. The four regions (A only, both, B only, neither) make it easy to organise the data and to apply the addition rule. Use a tree diagram when you have a sequential compound event — two or more stages happening in order, like drawing 2 marbles from a bag or flipping a coin three times. Each branch shows a probability at one stage, and you multiply along a path. Both tools work for "AND" / "OR" questions, but the wrong tool makes the problem much harder than it needs to be.

Marking: 1 mark for when to use Venn (overlapping sets); 1 mark for when to use tree (sequential compound events); 1 mark for a Venn example; 1 mark for a tree example with the word "compound event" used.