Mathematics • Year 10 • Unit 4 • Lesson 17
Venn Diagrams and Two-Way Tables — Skill Drill
Build fluency with Lesson 17's tools: Venn diagrams for intersections and unions; two-way tables for two categorical variables; and the addition rule P(A or B) = P(A) + P(B) − P(A and B). Make sure you do not confuse mutually exclusive with complementary.
1. I do — fully worked example
Read every step, including the small reason on the right of each line.
Problem. In a class of 30 students, 18 play basketball, 15 play soccer, and 8 play both. Build a Venn diagram and a two-way table, then find P(basketball or soccer).
Step 1 — Strip out the overlap first.
Basketball only = 18 − 8 = 10. Soccer only = 15 − 8 = 7.
Reason: students who do BOTH are counted in both the 18 and the 15. Remove them from the "only" regions.
Step 2 — Find the "neither" region.
Neither = 30 − (10 + 8 + 7) = 5.
Reason: total = (basketball only) + (both) + (soccer only) + (neither).
Step 3 — Build the two-way table.
Rows = basketball Yes/No, Cols = soccer Yes/No.
| Soccer Y | Soccer N | Total
BasY| 8 | 10 | 18
BasN| 7 | 5 | 12
Tot | 15 | 15 | 30
Reason: Lesson 17 Exam Tip — row totals and column totals must add up correctly. 18+12 = 30 ✓; 15+15 = 30 ✓.
Step 4 — Apply the addition rule.
P(B or S) = P(B) + P(S) − P(B and S) = 18/30 + 15/30 − 8/30 = 25/30 = 5/6.
Reason: Lesson 17 Remember card — "Addition rule: P(A or B) = P(A) + P(B) − P(A and B)."
Answer: P(basketball or soccer) = 5/6.
2. We do — fill in the missing steps
Same structure as Section 1, with the working faded. Fill the blanks. 4 marks
Problem. In a survey of 50 students, 30 like pizza, 25 like burgers, 12 like both. Find P(pizza or burger).
Step 1 — Subtract the overlap.
Pizza only = 30 − 12 = ______. Burger only = 25 − 12 = ______.
Step 2 — Find the "neither".
Neither = 50 − (18 + 12 + 13) = ______.
Step 3 — Addition rule.
P(P) = 30/50. P(B) = 25/50. P(P and B) = ______/50.
P(P or B) = 30/50 + 25/50 − ______/50 = ______/50 = ______ (simplified fraction).
Step 4 — Sanity check.
"Neither" + "Pizza or Burger" should equal the whole sample (50). Check: ______ + ______ = 50. ✓
3. You do — independent practice
Show working. Foundation = one calc each. Standard = full Venn / two-way table. Extension = complementary vs mutually exclusive trap.
Foundation — addition and complement
3.1 P(A) = 0.4 and P(A′) = ______ (the complement). 1 mark
3.2 Events A and B are mutually exclusive, P(A) = 0.25, P(B) = 0.35. Find P(A or B). 1 mark
3.3 P(A) = 0.6, P(B) = 0.5, P(A and B) = 0.3. Find P(A or B). 1 mark
3.4 A card is drawn from a standard deck. Are the events "heart" and "spade" mutually exclusive? One word answer with one-line reason. 1 mark
Standard — fill in a two-way table
3.5 A survey of 80 Year 10 students: 45 own a dog, 35 own a cat, 20 own both. Complete a two-way table with rows = Dog (Y/N), columns = Cat (Y/N). Verify your row and column totals. 3 marks
3.6 Using the same 80-student data, find P(dog or cat) and P(neither). 2 marks
Extension — the complementary vs mutually exclusive trap
3.7 A die is rolled. A = even number; B = a 3; C = odd number. For each pair, state whether they are mutually exclusive, complementary, or neither. Give a one-line reason. 3 marks
3.8 A friend says: "If two events are mutually exclusive, they must be complementary." Apply Lesson 17's misconception fix to show why this is wrong, using "even number" and "a 3" on a fair die as your example. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (pizza and burgers)
Step 1: Pizza only = 18. Burger only = 13.
Step 2: Neither = 50 − (18+12+13) = 7.
Step 3: P(P and B) = 12/50. P(P or B) = 30/50 + 25/50 − 12/50 = 43/50 = 0.86.
Step 4: 7 + 43 = 50 ✓.
3.1 — Complement
P(A′) = 1 − 0.4 = 0.6.
3.2 — Mutually exclusive
P(A or B) = 0.25 + 0.35 = 0.60.
3.3 — Addition rule with overlap
P(A or B) = 0.6 + 0.5 − 0.3 = 0.8.
3.4 — Heart and spade
Yes, mutually exclusive. A card cannot be both a heart and a spade — they are different suits.
3.5 — Two-way table for 80 students
Dog only = 25, Cat only = 15, both = 20, neither = 80 − 60 = 20.
| Cat Y | Cat N | Total
Dog Y | 20 | 25 | 45
Dog N | 15 | 20 | 35
Total | 35 | 45 | 80
Row totals 45+35 = 80 ✓; column totals 35+45 = 80 ✓.
3.6 — Dog or cat / neither
P(dog or cat) = 60/80 = 3/4. P(neither) = 20/80 = 1/4. (Together they sum to 1 ✓.)
3.7 — Die pairs
A (even) & B (a 3): mutually exclusive. 3 is odd, no overlap.
A (even) & C (odd): complementary. Mutually exclusive AND together cover all outcomes.
B (a 3) & C (odd): neither. 3 is odd, so a roll of 3 is in BOTH events — not mutually exclusive.
3.8 — Mutually exclusive ≠ complementary
"Even" and "a 3" are mutually exclusive — no roll is both — but they are not complementary: a roll of 1 or 5 is neither even nor a 3. Complementary events not only cannot both occur, they must together cover the whole sample space. Lesson 17 misconception fix: "Mutually exclusive events cannot both occur, but neither might occur — complementary events guarantee one must occur."