Mathematics • Year 7 • Unit 4 • Lesson 18
Sample Space and Tree Diagrams — Real World
Apply tree diagrams, grids and the multiplication principle to real planning problems: a school uniform shop, a takeaway menu, a sports tournament, a quiz combo, and a school PIN-code lock.
1. Word problems
Use the multiplication principle (or a tree diagram or grid) and show your working.
1.1 — School uniform shop. The shop sells the uniform in 3 polo colours (navy, sky blue, white), 2 short styles (sport or chino), and 4 size codes (S, M, L, XL). (a) How many distinct uniform packages are possible? (b) If the colour and short style is chosen at random, find the probability of "navy polo and chino shorts". 3 marks
1.2 — Takeaway menu. A burger shop offers 5 burger choices, 3 side choices, and 4 drink choices. (a) How many "meal deals" (one burger, one side, one drink) are possible? (b) Two of the burgers are vegetarian. If a customer picks at random, find P(vegetarian meal deal). 3 marks
1.3 — Sports tournament draw. Three Year 7 sport groups (A, B, C) play each other once in a round-robin tournament. (a) Draw a tree diagram for "team 1 vs team 2" of all the matches. (b) How many distinct matches are played? (c) Out of all the ordered pairs (team 1, team 2) in your tree, how many have team A playing? 3 marks
1.4 — Quiz combo. A Year 7 trivia quiz has 4 categories (Science, Sport, Music, History). Two categories are chosen, one for round 1 and a different one for round 2 (order matters). (a) Draw a tree diagram or grid and count the outcomes. (b) Find P(round 1 = Science). 3 marks
1.5 — Classroom PIN-code lock. The science lab door uses a 4-digit PIN where every digit is from 0–9 and repeats are allowed. (a) How many PINs are possible? (b) Find the probability that a randomly chosen PIN starts with the digit 7. 3 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A Year 7 student says: "Flipping two coins has 3 possible outcomes — both heads, both tails, or one of each — so P(both heads) = 1/3." In your own words, explain (i) why this is wrong, (ii) what the correct sample space is, and (iii) what role a tree diagram (or grid) plays in catching this mistake.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Uniform shop
(a) Total packages = 3 × 2 × 4 = 24.
(b) Colour-and-style pairs = 3 × 2 = 6. "Navy polo and chino" is 1 favourable pair. P = 1/6.
1.2 — Takeaway menu
(a) Meal deals = 5 × 3 × 4 = 60.
(b) Veg meal deals = 2 × 3 × 4 = 24 out of 60. P(veg) = 24/60 = 2/5 = 0.4.
1.3 — Sports tournament
(a) Tree pairs (team 1 → team 2): A→B, A→C, B→A, B→C, C→A, C→B (6 ordered pairs).
(b) Unique matches = 3 (A-B, A-C, B-C — order does not matter).
(c) Pairs with team A appearing in either slot: A→B, A→C, B→A, C→A = 4 out of 6 ordered pairs.
1.4 — Quiz combo
(a) Round 1 = 4 options; Round 2 = 3 remaining options. Total = 4 × 3 = 12 outcomes.
(b) Round 1 = Science gives 1 × 3 = 3 outcomes out of 12. P(Round 1 = Science) = 3/12 = 1/4.
1.5 — PIN-code lock
(a) Total PINs = 10 × 10 × 10 × 10 = 10 000.
(b) PINs starting with 7 = 1 × 10 × 10 × 10 = 1000. P(starts with 7) = 1000/10 000 = 1/10 = 0.1.
2.1 — Explain your thinking (sample response)
(i) The student has lumped HT (heads first, then tails) and TH (tails first, then heads) into one "one of each" outcome. These are actually two different paths in the experiment, so collapsing them undercounts the sample space.
(ii) The correct sample space has 2 × 2 = 4 equally likely outcomes: {HH, HT, TH, TT}. So P(both heads) = 1/4, not 1/3.
(iii) A tree diagram makes the mistake impossible — you read every full path from left to right, and HT and TH come out as two distinct paths. A grid does the same: HT is in one cell, TH is in another. The multiplication principle (2 × 2 = 4) gives the correct count even without a picture.
Marking: 1 for identifying the HT vs TH error, 1 for the correct sample space, 1 for explaining the role of the tree/grid, 1 for clear sentences.