Mathematics • Year 7 • Unit 4 • Lesson 14
Probability — Mixed Challenge
Bring together sample-space listing, the probability formula, the complement rule, and the rule that P is always between 0 and 1. Spot a probability mistake, then design your own probability puzzle.
1. Mixed problems
Each question uses the probability formula or the complement rule. Show your working. 2 marks each
1.1 A standard die is rolled. Find P(rolling a number less than 4) as a fraction and a decimal.
1.2 A bag has 7 red, 5 blue and 8 green marbles. Find P(green).
1.3 If P(passing the driving test on the first try) = 0.45, what is P(failing on the first try)?
1.4 A letter is chosen from the word "MATHEMATICS" (11 letters total). Find P(choosing the letter M). Then find P(choosing a vowel).
1.5 An 8-section spinner is numbered 1–8. Find P(spinning an even prime). (Hint: list the primes 2, 3, 5, 7 — which are even?)
1.6 A student writes P(picking a red ball from a bag) = 6/4. Explain in one sentence why this is impossible and suggest what they likely got wrong.
2. Find the mistake
A Year 7 student wrote the following solution. Exactly one statement contains an error. 3 marks
Problem: A bag contains 4 red, 3 blue and 5 green counters. Find P(red) and P(not red).
Line 1: Total counters = 4 + 3 + 5 = 12.
Line 2: Favourable (red) = 4. So P(red) = 4/12 = 1/3.
Line 3: P(not red) = 1 − P(red) = 1 − 1/3 = 3/3 = 1.
Line 4: Check: 3 blue + 5 green = 8 counters that are not red. P(not red) should be 8/12 = 2/3.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected calculation.
Stuck? Compute 1 − 1/3 carefully. The correct denominator stays the same.3. Open-ended challenge — design a fair game
This question has many correct answers. Show your work clearly. 4 marks
3.1 You are designing a school fete game using a spinner with 12 equal sections. You need to set it up so that exactly:
- P(big prize) = 1/12,
- P(small prize) = 1/3,
- P(try again) = 1/4,
- P(no prize) accounts for the rest.
For each outcome: (i) state how many sections should be labelled for that outcome, (ii) verify that all four probabilities sum to 1, (iii) calculate P(no prize) using the complement rule, (iv) state whether this is a "fair" game from a player's perspective.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Less than 4 on a die
Favourable {1, 2, 3} = 3. P = 3/6 = 1/2 = 0.5.
1.2 — P(green)
Total = 7 + 5 + 8 = 20. P(green) = 8/20 = 2/5 = 0.4.
1.3 — Driving test
P(fail) = 1 − 0.45 = 0.55.
1.4 — MATHEMATICS
M appears twice in MATHEMATICS, so P(M) = 2/11.
Vowels: A, E, A, I = 4 vowels. P(vowel) = 4/11.
1.5 — Even prime on spinner
Primes in 1–8 = {2, 3, 5, 7}. The only EVEN prime is 2. So 1 favourable out of 8 total. P = 1/8 = 0.125.
1.6 — P = 6/4 is impossible
P = 6/4 = 1.5 is greater than 1, which is impossible. The student must have written the favourable count (6) over a wrong denominator (4 instead of the true total) — perhaps they used the number of one colour as the denominator instead of the total number of balls in the bag.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) 1 − 1/3 ≠ 3/3 = 1. Correctly: 1 − 1/3 = 3/3 − 1/3 = 2/3. The student wrote 3/3 (the whole) instead of subtracting properly.
(c) Corrected: P(not red) = 1 − 1/3 = 2/3. This matches Line 4's check (8/12 = 2/3).
3 — Fete game (sample design)
(i) Sections out of 12:
Big prize: 1/12 → 1 section.
Small prize: 1/3 = 4/12 → 4 sections.
Try again: 1/4 = 3/12 → 3 sections.
No prize: remaining = 12 − 1 − 4 − 3 = 4 sections.
(ii) Check: 1/12 + 4/12 + 3/12 + 4/12 = 12/12 = 1 ✓.
(iii) P(no prize) using complement = 1 − P(any prize or try again) = 1 − (1/12 + 4/12 + 3/12) = 1 − 8/12 = 4/12 = 1/3. Matches.
(iv) From a player's perspective: P(big prize) is very low (1/12 ≈ 8%) and P(no prize) is 1/3. The game is mathematically defined, but a player who wants a "big prize" should know they have only about an 8% chance per spin.
Marking: 1 for each section calculation; 1 for verifying they sum to 1; 1 for P(no prize) with complement; 1 for one-sentence judgement.