Mathematics • Year 10 • Unit 4 • Lesson 17
Venn Diagrams and Two-Way Tables — Mixed Challenge
Pull Lesson 17 together: Venn diagrams, two-way tables, the addition rule with overlap, and the difference between mutually exclusive, complementary and independent events. Spot a Year 10 mistake and design a survey of your own.
1. Mixed problems — pick the right tool
Decide whether the question wants the addition rule, the complement, a Venn diagram or a two-way table before you start. 3 marks each
1.1 In a deck of cards, find P(face card OR heart). (Face cards are J, Q, K — 12 in total; 3 of them are hearts.)
1.2 A die is rolled. P(rolling a 3) = 1/6. Find the probability of NOT rolling a 3, naming the rule you used.
1.3 A survey of 120 students records two categorical variables: own a pet (Y/N) and own a bicycle (Y/N). 80 own a pet, 70 own a bicycle, 45 own both. Build the two-way table.
1.4 Using the same survey, find P(own a pet OR own a bicycle), P(own a pet but NOT a bicycle), and P(neither).
1.5 Apply the Lesson 17 HSC Note: for mutually exclusive events, the addition rule simplifies. Show this in symbols, and explain in one sentence why it does NOT work for events that can both occur.
1.6 Two events A and B have P(A) = 0.5, P(B) = 0.4 and are independent. Use Lesson 16 (multiplication rule for independence) to find P(A and B), then use Lesson 17's addition rule to find P(A or B).
2. Find the mistake
A Year 10 student tackles this: "In a class of 30 students, 18 play sport, 15 play music, 10 do both. Find P(sport or music)." Exactly one line is wrong. 3 marks
Student's working:
Line 1: n(sport) = 18. n(music) = 15. n(both) = 10.
Line 2: "Sport or music" means add: n(sport or music) = 18 + 15 = 33.
Line 3: P(sport or music) = 33/30 = 1.1.
Line 4: So P(sport or music) = 1.1.
(a) Which line is wrong?
(b) Explain why, naming the Lesson 17 misconception in your answer.
(c) Write the corrected working and the correct P(sport or music).
Stuck? A probability cannot exceed 1. That is the giveaway that you have double-counted.3. Open-ended challenge — design your own survey
Many valid answers. Follow every rule. 4 marks
3.1 Design a school survey of exactly 100 students that records two categorical variables of your choice (eg. plays sport / plays music, owns a pet / has a sibling, etc.). Choose your own numbers, but you must:
- have an overlap that is at least 10 students,
- have a "neither" region that is at least 5 students,
- have the two events not be mutually exclusive (i.e. there are people in both).
Write up:
(i) the survey scenario in one sentence,
(ii) a Venn diagram with all four regions filled in,
(iii) the matching two-way table with row and column totals,
(iv) P(A), P(B), P(A and B), P(A or B) — show the addition-rule calculation in full.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Face card or heart
P(face) = 12/52. P(heart) = 13/52. Overlap (face card AND heart) = J, Q, K of hearts = 3/52.
P(face or heart) = 12/52 + 13/52 − 3/52 = 22/52 = 11/26.
1.2 — Complement rule
P(not 3) = 1 − P(3) = 1 − 1/6 = 5/6. Rule used: complementary events, P(A) + P(A′) = 1.
1.3 — Two-way table
Pet only = 35, both = 45, bike only = 25, neither = 120 − 105 = 15.
| Bike Y | Bike N | Total
Pet Y | 45 | 35 | 80
Pet N | 25 | 15 | 40
Tot | 70 | 50 | 120 ✓
1.4 — Probabilities from the table
P(pet or bike) = 105/120 = 7/8.
P(pet but not bike) = 35/120 = 7/24.
P(neither) = 15/120 = 1/8.
1.5 — Simplified addition rule
For mutually exclusive A and B, P(A and B) = 0, so P(A or B) = P(A) + P(B) − 0 = P(A) + P(B). If the events can both occur, P(A and B) ≠ 0 and skipping the subtraction double-counts the overlap, often giving a probability greater than 1.
1.6 — Independent A and B
P(A and B) = 0.5 × 0.4 = 0.2.
P(A or B) = 0.5 + 0.4 − 0.2 = 0.7.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student added n(sport) and n(music) without subtracting the 10 students who play both. Lesson 17 misconception fix: "n(A or B) = n(A) + n(B) − n(A and B). The intersection is subtracted because it is counted twice." A probability of 1.1 (Line 3) is the giveaway.
(c) Corrected: n(sport or music) = 18 + 15 − 10 = 23. P(sport or music) = 23/30.
3 — Open-ended challenge (sample solution)
Scenario: 100 Year 10 students surveyed on owning a phone (A) and owning a tablet (B). 70 own a phone, 40 own a tablet, 25 own both.
Venn: phone only = 45, both = 25, tablet only = 15, neither = 100 − 85 = 15.
Two-way table:
| Tab Y | Tab N | Total
Ph Y | 25 | 45 | 70
Ph N | 15 | 15 | 30
Tot | 40 | 60 | 100 ✓
Probabilities: P(A) = 70/100 = 0.7. P(B) = 40/100 = 0.4. P(A and B) = 25/100 = 0.25. P(A or B) = 0.7 + 0.4 − 0.25 = 0.85.
Marking: 1 mark — scenario follows the three rules; 1 — Venn diagram with all four regions correct; 1 — two-way table with valid row/column totals; 1 — addition-rule calculation shown with final answer.