Mathematics • Year 8 • Unit 4 • Lesson 15
The Probability Scale — Mixed Challenge
Pull together calculating, converting (fraction/decimal/percentage), labelling on the scale, and ordering events by likelihood. Six mixed problems, one "find the mistake", and one open-ended challenge where YOU design a scale with five labelled events.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 15. Show working. 3 marks each
1.1 Convert each to its other two representations (fraction, decimal, percentage):
(a) 3/5 (b) 0.07 (c) 92%
1.2 Label each event on the scale (impossible / unlikely / even chance / likely / certain):
(a) Rolling an even number on a die (P = ?).
(b) Drawing a heart from a standard deck (P = ?).
(c) Drawing a card that is not a 13 (e.g. not a King) — express as fraction first.
1.3 Order these five events from LEAST likely to MOST likely. Calculate each probability first.
Event 1: Rolling a 6 on a standard die.
Event 2: Flipping at least one head in 2 coin flips.
Event 3: Drawing a face card (J, Q, K) from a standard deck.
Event 4: Rolling an odd number on a die.
Event 5: Drawing the Ace of Spades specifically from a standard deck.
1.4 Compare 7/15 and 11/25. (a) Which is greater? (b) Convert both to decimals (3 d.p.). (c) Label each on the scale.
1.5 A spinner has 20 equal sectors: 8 are red, 6 are blue, 5 are green, 1 is gold. (a) Calculate P(red), P(blue), P(green), P(gold) as percentages. (b) Label each on the scale. (c) Verify the four probabilities add to 1 (or 100%).
1.6 A friend claims: "The probability of me winning the lottery is 1 in 14 million, which is the same as zero — it's impossible." Use the probability scale to respond: (a) what label does P = 1/14 000 000 actually get? (b) Why is it WRONG to call this impossible?
2. Find the mistake
Another student tried to label and convert a probability. Exactly one line below contains a mistake. Spot it, explain why, and correct it. 3 marks
Problem: A bag has 9 white and 1 black marble. Convert P(black) to a decimal and percentage, and label it on the scale.
Line 1: Total marbles = 9 + 1 = 10.
Line 2: P(black) = 1 ÷ 10 = 1/10.
Line 3: As a decimal: 1/10 = 0.10.
Line 4: As a percentage: 0.10 × 100 = 100%.
Line 5: Label on scale: 100% is "certain".
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected working, the correct percentage, and the correct label.
Stuck? Decimal → percentage: 0.10 × 100 = 10%, not 100%. The student multiplied incorrectly.3. Open-ended challenge — design a probability scale
This question has many valid answers. 4 marks
3.1 Your job: design a probability scale showing FIVE different events from your own life — one for each of the five labels (impossible, unlikely, even chance, likely, certain).
(i) Choose a real-feeling event for EACH of the five labels (e.g. "rolling a 7 on a die", "winning the school lottery", "flipping a coin and getting tails", "passing tomorrow's spelling test if I study", "the sun rising tomorrow").
(ii) Estimate (or calculate where possible) a probability value for each event as a fraction OR decimal OR percentage.
(iii) Order the five events from least to most likely.
(iv) Draw (or describe in words) a probability number line from 0 to 1 with all five events placed on it.
(v) Pick one "unlikely" or "likely" event and explain in one sentence why it is NOT at the extreme (0 or 1) — i.e. why there is still uncertainty.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Convert
(a) 3/5 = 0.6 = 60%. (b) 0.07 = 7/100 = 7%. (c) 92% = 0.92 = 23/25 (or 92/100).
1.2 — Label on the scale
(a) Even number on die: P = 3/6 = 1/2 = 0.5 → even chance.
(b) Heart from deck: P = 13/52 = 1/4 = 0.25 → unlikely.
(c) Not a King: P = 48/52 = 12/13 ≈ 0.923 → likely (very close to certain).
1.3 — Order five events
E1 = 1/6 ≈ 0.167. E2 = 3/4 = 0.75 (sample space HH, HT, TH, TT — 3 have at least one head). E3 = 12/52 ≈ 0.231. E4 = 3/6 = 0.5. E5 = 1/52 ≈ 0.019.
Order (least → most likely): E5 (0.019) < E1 (0.167) < E3 (0.231) < E4 (0.5) < E2 (0.75).
1.4 — Compare 7/15 vs 11/25
(a) 7/15 ≈ 0.467; 11/25 = 0.440. So 7/15 is greater.
(b) Decimals shown above.
(c) Both 0.467 and 0.440 are between 0 and 0.5, so both are unlikely (very close to even chance).
1.5 — 20-sector spinner
(a) P(red) = 8/20 = 40%. P(blue) = 6/20 = 30%. P(green) = 5/20 = 25%. P(gold) = 1/20 = 5%.
(b) Red (40%) = unlikely. Blue (30%) = unlikely. Green (25%) = unlikely. Gold (5%) = unlikely (very close to impossible).
(c) Sum = 40 + 30 + 25 + 5 = 100% ✓ (or 8 + 6 + 5 + 1 = 20 sectors out of 20).
1.6 — Lottery friend
(a) P = 1/14 000 000 ≈ 0.00000007 — sits very close to 0 on the scale. Label: unlikely (specifically, extremely unlikely).
(b) It is wrong to call it "impossible" because impossibility means P = exactly 0. A very small probability is still positive, so the event CAN happen — people DO occasionally win the lottery. The scale distinguishes "tiny but possible" from "absolutely cannot happen".
2 — Find the mistake
(a) The mistake is on Line 4.
(b) To convert decimal to percentage, multiply by 100. 0.10 × 100 = 10%, not 100%. The student misread the multiplication, possibly confusing 0.10 with 1.00.
(c) Corrected: 0.10 × 100 = 10%. Label: P = 10% is between 0 and 50%, so it is unlikely, not certain. (Drawing a black marble from a bag of 9 white and 1 black is obviously unlikely.)
3 — Open-ended challenge (sample solution)
(i) Five events:
Impossible — Rolling a 7 on a single standard die (P = 0).
Unlikely — Getting picked first in a class draw of 30 (P = 1/30 ≈ 0.033).
Even chance — Flipping heads on a fair coin (P = 0.5).
Likely — Passing my spelling test if I study tonight (P ≈ 0.85).
Certain — The sun rising tomorrow (P = 1).
(iii) Order (least to most): Roll 7 (0) < Picked first (0.033) < Heads (0.5) < Pass spelling (0.85) < Sun rises (1).
(iv) Number line: 0 ——— Impossible(roll 7) ········ Picked first(0.033) ································· Heads(0.5) ··············· Spelling pass(0.85) ······ Sun rises(1) ——— 1.
(v) Why not extreme? The spelling test pass is "likely" but NOT certain because I might still forget a word under pressure, or new words might appear that I didn't revise — so there is real uncertainty.
Marking: 1 mark for five plausible events (one per label). 1 mark for sensible probability values. 1 mark for correct ordering on the scale. 1 mark for an honest sentence explaining why a "likely" or "unlikely" event isn't at 0 or 1.