Mathematics Advanced • Year 12 • Module 5 • Lesson 1
Introduction to Probability
Build procedural fluency in sample spaces, Venn-diagram regions, the addition rule and complement counting.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete each definition with the correct word or symbol:
The set of all possible outcomes of an experiment is the ____________ space, written ______.
For equally likely outcomes, P(A) = ______ / ______.
The complement of A is written ______, and satisfies P(A) + P(A′) = ______.
Q1.2 State the symbol that means each of the following set operations:
"A or B (at least one)" → ______ "A and B (both)" → ______ "not A" → ______
Write the addition rule in full: P(A ∪ B) = _________________________________.
Q1.3 Two events A and B are mutually exclusive. State the value of P(A ∩ B) and the simplified form of P(A ∪ B). P(A ∩ B) = ______ P(A ∪ B) = _________________
2. Worked example, P(B ∪ T) for a sport survey
Follow each step. Reasons are in italics on the right.
Problem. In a class of 30 students, 18 play basketball (B), 15 play tennis (T) and 8 play both. Find P(B), P(T), P(B ∩ T) and P(B ∪ T).
Step 1, Identify the sample space size.
n(S) = 30
Reason: every student is one equally likely outcome.
Step 2, Single-event probabilities.
P(B) = 18/30 = 3/5 P(T) = 15/30 = 1/2
Reason: P(A) = n(A)/n(S).
Step 3, Intersection.
P(B ∩ T) = 8/30 = 4/15
Reason: 8 students belong to both events; they sit in the overlap of the Venn diagram.
Step 4, Apply the addition rule.
P(B ∪ T) = P(B) + P(T) − P(B ∩ T)
= 18/30 + 15/30 − 8/30 = 25/30 = 5/6
Reason: adding 18 + 15 double-counts the 8 students in the overlap, so we subtract them once.
Conclusion. P(B ∪ T) = 5/6. The Venn regions are: B only = 10, T only = 7, both = 8, neither = 5 (total 30). ✓
3. Faded example, fill in the missing steps
In a group of 40 students, 22 study Biology (B), 18 study Chemistry (C), and 10 study both. Find P(B ∪ C). 4 marks
Step 1, Sample space. n(S) = ______
Step 2, Single events. P(B) = ______ / 40 = ______ P(C) = ______ / 40 = ______
Step 3, Intersection. P(B ∩ C) = ______ / 40 = ______
Step 4, Addition rule.
P(B ∪ C) = ______ + ______ − ______ = ______ / 40 = ______
Step 5, Venn check. B only = ______, C only = ______, both = ______, neither = ______ Total = 40 ✓
4. Graduated practice, calculate the probability
Show the substitution and final answer as a simplified fraction or 3-decimal-place decimal. Assume fair / unbiased equipment unless stated.
Foundation, single-event probabilities (4 questions)
| Q | Question | Working (one line) | Probability |
|---|---|---|---|
| 4.1 1 | Roll one fair die. P(rolling a 4). | ||
| 4.2 1 | Draw one card from a standard 52-card deck. P(heart). | ||
| 4.3 1 | Roll one die. P(even number). | ||
| 4.4 1 | Flip two fair coins. P(at least one head). List the sample space first. |
Standard, typical HSC difficulty (6 questions)
Show one line of working in each box.
4.5 Roll two fair dice. Find P(sum = 7). Hint: list the favourable ordered pairs. 2 marks
4.6 Given P(A) = 0.4, P(B) = 0.3 and P(A ∩ B) = 0.15, find P(A ∪ B). 2 marks
4.7 A bag holds 5 red, 3 blue and 2 green marbles. One marble is drawn. Find P(red ∪ blue). 2 marks
4.8 A die is rolled. Find P(rolling a number that is even or greater than 4). Use the addition rule and watch for overlap. 2 marks
4.9 A fair coin is flipped four times. Use the complement to find P(at least one tail). 2 marks
4.10 In a survey of 80 people: 45 own a smartphone, 50 own a laptop and 30 own both. Find P(owns smartphone or laptop). 2 marks
Extension, reasoning beyond procedure (2 questions)
4.11 Two events satisfy P(A) = 0.6 and P(B) = 0.5. Find the smallest and largest possible values of P(A ∪ B), and describe geometrically (in terms of a Venn diagram) what each extreme corresponds to. 3 marks
4.12 Three events A, B, C divide the sample space into 8 disjoint regions. Sketch a 3-circle Venn diagram and label all 8 regions in set notation (e.g. A ∩ B ∩ C′). 3 marks
5. Self-check the easy 3
Tick the first three once you've verified your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, Definitions
Sample space, written S. P(A) = n(A) / n(S). Complement A′; P(A) + P(A′) = 1.
Q1.2, Symbols and addition rule
"A or B" → ∪; "A and B" → ∩; "not A" → A′. P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Q1.3, Mutually exclusive
P(A ∩ B) = 0; P(A ∪ B) = P(A) + P(B) (no overlap to subtract).
Q3, Faded example (Biology / Chemistry)
Step 1: n(S) = 40.
Step 2: P(B) = 22/40 = 11/20; P(C) = 18/40 = 9/20.
Step 3: P(B ∩ C) = 10/40 = 1/4.
Step 4: P(B ∪ C) = 22/40 + 18/40 − 10/40 = 30/40 = 3/4.
Step 5: B only = 12, C only = 8, both = 10, neither = 10. Total = 40 ✓.
Q4.1, P(rolling a 4)
n(A) = 1, n(S) = 6, so P = 1/6.
Q4.2, P(heart)
13 hearts in 52 cards: P = 13/52 = 1/4.
Q4.3, P(even)
Favourable outcomes {2, 4, 6}, so P = 3/6 = 1/2.
Q4.4, P(at least one head, two coins)
S = {HH, HT, TH, TT}, n(S) = 4. Favourable: {HH, HT, TH}, so P = 3/4. (Equivalently, 1 − P(TT) = 1 − 1/4 = 3/4.)
Q4.5, P(sum = 7) on two dice
Favourable ordered pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), 6 pairs. P = 6/36 = 1/6.
Q4.6, Addition rule
P(A ∪ B) = 0.4 + 0.3 − 0.15 = 0.55.
Q4.7, P(red ∪ blue) marbles
Red and blue are mutually exclusive (one marble cannot be two colours), so P = 5/10 + 3/10 = 8/10 = 4/5.
Q4.8, P(even or > 4) on a die
Even = {2, 4, 6} so P(even) = 3/6. >4 = {5, 6} so P(>4) = 2/6. Overlap = {6}, P(overlap) = 1/6.
P(even ∪ >4) = 3/6 + 2/6 − 1/6 = 4/6 = 2/3.
Q4.9, P(at least one tail in 4 flips)
P(no tails) = P(HHHH) = (1/2)⁴ = 1/16. So P(at least one tail) = 1 − 1/16 = 15/16.
Q4.10, Smartphone or laptop
P = 45/80 + 50/80 − 30/80 = 65/80 = 13/16.
Q4.11, Bounds on P(A ∪ B)
Largest: if A and B are mutually exclusive (no overlap), P(A ∪ B) = 0.6 + 0.5 = 1.1 but probabilities cannot exceed 1, so the largest feasible value is 1, with overlap forced to be at least 0.1. Smallest: if one event is contained inside the other, P(A ∪ B) = max(P(A), P(B)) = 0.6; geometrically, the smaller circle (B) sits entirely inside the larger (A), so the union equals the larger circle.
Q4.12, Eight regions of a 3-circle Venn
(i) A ∩ B′ ∩ C′ (ii) A′ ∩ B ∩ C′ (iii) A′ ∩ B′ ∩ C (iv) A ∩ B ∩ C′ (v) A ∩ B′ ∩ C (vi) A′ ∩ B ∩ C (vii) A ∩ B ∩ C (viii) A′ ∩ B′ ∩ C′ (outside all three).