Mathematics • Year 8 • Unit 4 • Lesson 20
Probability Review
Build fluency across every Unit 4 probability tool: the formula and scale, complement rule, counting principle, Venn diagrams, tree diagrams, and experimental probability. One worked example, one guided example, then eight independent problems mixing all of these.
1. I do — fully worked example (multi-tool review)
One problem that exercises three tools: a Venn diagram, the addition rule, and the complement. Watch how each step uses a different idea from the unit.
Problem. Of 50 students in a year group, 28 like tea (T), 22 like coffee (C), and 10 like both. (a) Fill in the four Venn regions. (b) Find P(T ∪ C) directly from regions. (c) Verify with the addition rule. (d) Find P(neither).
Step 1 — Place the intersection.
n(T ∩ C) = 10 → write 10 in the overlap
Reason: always start with the intersection (Lesson 17).
Step 2 — Find the "only" regions.
T only = 28 − 10 = 18 C only = 22 − 10 = 12
Step 3 — Neither.
Neither = 50 − 18 − 10 − 12 = 10. Check: 18+10+12+10 = 50 ✓
Step 4 — Direct count of P(T ∪ C).
n(T ∪ C) = 18 + 10 + 12 = 40 → P(T ∪ C) = 40/50 = 4/5
Step 5 — Verify with the addition rule.
P(T) + P(C) − P(T ∩ C) = 28/50 + 22/50 − 10/50 = 40/50 = 4/5 ✓
Step 6 — P(neither) using the complement.
P(neither) = 1 − P(T ∪ C) = 1 − 4/5 = 1/5 (matches 10/50 ✓)
Answer: Regions are 18, 10, 12, 10; P(T ∪ C) = 4/5; P(neither) = 1/5.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. Two coins are flipped. (a) Find |S| using the counting principle. (b) Find P(exactly one head). (c) Verify endpoint probabilities on a tree sum to 1.
Step 1 — Counting principle:
|S| = ______ × ______ = ______
Step 2 — List the four outcomes:
______, ______, ______, ______
Step 3 — Identify favourable for "exactly one H":
______ and ______ (NOT HH or TT)
Step 4 — Each endpoint probability on tree:
1/2 × 1/2 = ______ for each path
Step 5 — Sum-to-1 check:
______ + ______ + ______ + ______ = ______ ✓
Step 6 — P(exactly one H):
P = ______ + ______ = ______
3. You do — independent practice (mixed)
Show working under each problem. These mix every tool from the unit: scale, complement, counting principle, Venn, tree, and experimental.
Foundation — formula and scale
3.1 A bag has 4 red, 3 blue, and 2 green marbles. Find P(red). State whether the probability is closer to 0, 1/2, or 1. 1 mark
3.2 A fair die is rolled. Use the complement rule to find P(NOT rolling a 6). 1 mark
3.3 A coin is flipped and a 5-sector spinner (1-5) is spun. State |S| using the counting principle. 1 mark
3.4 n(ξ) = 40, A only = 15, n(A ∩ B) = 5, B only = 12. Find n(neither). 1 mark
Standard — combine ideas
3.5 Two dice are rolled. Find P(sum > 8) by listing all favourable ordered pairs. Simplify. 2 marks
3.6 A bag has 2 R and 3 B marbles. Two are drawn with replacement. Use a tree diagram to find P(both same colour). 2 marks
Extension — experimental and Venn
3.7 A spinner is spun 120 times and lands on red 48 times. (a) Find experimental P(red). (b) If theoretical P(red) = 0.4, is this consistent with a fair spinner? 2 marks
3.8 Of 40 students: 22 play sport (S), 18 play an instrument (I), 8 do both. (a) Find n(neither). (b) Find P(plays sport only). 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (two coins)
Step 1: |S| = 2 × 2 = 4. Step 2: HH, HT, TH, TT. Step 3: HT and TH. Step 4: Each endpoint = 1/4. Step 5: 1/4 + 1/4 + 1/4 + 1/4 = 1 ✓. Step 6: P(exactly one H) = 1/4 + 1/4 = 1/2.
3.1 — Marbles in bag
P(red) = 4/9 ≈ 0.44. Closest to 1/2.
3.2 — Complement rule
P(NOT a 6) = 1 − P(6) = 1 − 1/6 = 5/6.
3.3 — Coin and spinner
|S| = 2 × 5 = 10.
3.4 — Venn neither
Neither = 40 − 15 − 5 − 12 = 8.
3.5 — Two dice, sum > 8
Sum 9: (3,6), (4,5), (5,4), (6,3) → 4.
Sum 10: (4,6), (5,5), (6,4) → 3.
Sum 11: (5,6), (6,5) → 2.
Sum 12: (6,6) → 1.
Favourable = 4 + 3 + 2 + 1 = 10. P(sum > 8) = 10/36 = 5/18.
3.6 — Tree, with replacement, same colour
P(R) = 2/5, P(B) = 3/5 at every draw.
P(RR) = 4/25; P(BB) = 9/25.
P(same) = 4/25 + 9/25 = 13/25.
3.7 — Spinner experimental
(a) P(red) ≈ 48/120 = 2/5 = 0.40.
(b) Matches theoretical exactly — fully consistent with a fair spinner.
3.8 — Sport and instrument Venn
S only = 22 − 8 = 14; I only = 18 − 8 = 10.
(a) Neither = 40 − 14 − 8 − 10 = 8.
(b) P(sport only) = 14/40 = 7/20.