Mathematics • Year 10 • Unit 4 • Lesson 15
Introduction to Probability — Skill Drill
Build fluency with Lesson 15's core formula: P(event) = (favourable outcomes) / (total outcomes), where the answer always sits in [0, 1]. Practise sample spaces, the complement rule P(A′) = 1 − P(A), and the lesson's tree-diagram trap (HH, HT, TH, TT are four outcomes, not three).
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. A bag contains 4 red, 5 blue and 1 green marble. A marble is drawn at random. Find:
(a) P(red), (b) P(not blue), (c) P(red or green).
Step 1 — Sample space + total.
Total marbles = 4 + 5 + 1 = 10. Sample space size = 10.
Reason: Lesson 15 key term — "sample space: the set of all possible outcomes".
Step 2 — P(red) using the basic formula.
Favourable = 4 (red). P(red) = 4/10 = 2/5 = 0.4 = 40 %.
Reason: P(event) = favourable / total — Lesson 15 callout.
Step 3 — P(not blue) using the complement rule.
P(blue) = 5/10 = 1/2. P(not blue) = 1 − 1/2 = 1/2 = 0.5.
Reason: Lesson 15 key term — complementary event: P(A′) = 1 − P(A).
Step 4 — P(red or green) by counting favourable outcomes.
Favourable = 4 red + 1 green = 5. P(red or green) = 5/10 = 1/2 = 0.5.
Reason: red and green are mutually exclusive, so just add the favourable counts.
Answer: P(red) = 0.4, P(not blue) = 0.5, P(red or green) = 0.5. All values lie in [0, 1]. ✓
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. A fair six-sided die is rolled. Find:
(a) P(even), (b) P(less than 3), (c) P(not a 6).
Step 1 — Sample space. S = {________________}. Total outcomes = ____.
Step 2 — P(even).
Favourable (even) = {________________} → count ____.
P(even) = ____ / ____ = ____.
Step 3 — P(less than 3).
Favourable = {________________} → count ____. P(<3) = ____ / ____ = ____.
Step 4 — P(not a 6) using the complement rule.
P(6) = ____ / ____ = ____. P(not 6) = 1 − ____ = ____.
Check: Each probability sits between ____ and ____.
3. You do — independent practice
For each question, state the sample space size, identify the favourable outcomes and find the probability as a fraction (and decimal where asked). Foundation = simple events. Standard = compound events. Extension = lesson trap and complement rule.
Foundation — simple events
3.1 A coin is flipped once. Find P(heads). 1 mark
3.2 A card is drawn from a standard 52-card deck. Find P(heart). 1 mark
3.3 A spinner is divided into 8 equal sections numbered 1–8. Find P(spinning an odd number). 1 mark
3.4 Letters A, B, C, D, E, F are written on cards and one is drawn at random. Find P(vowel). 1 mark
Standard — compound and complement
3.5 A coin is flipped TWICE. Write the sample space, then find: (a) P(two heads), (b) P(at least one tail). 2 marks
3.6 P(rain tomorrow) = 0.3. Use the complement rule to find P(no rain tomorrow). Then express as a percentage. 2 marks
Extension — lesson misconception traps
3.7 Lesson 15 misconception card says students often think flipping a coin TWICE gives 3 outcomes (HH, HT, TT). Explain in one sentence why this is wrong and write the correct sample space. Then find P(exactly one head). 3 marks
3.8 A student claims P(winning lotto) = 1.5 because "it would be amazing". Use Lesson 15's key term to explain why this is impossible, and state the correct maximum value any probability can take. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (die)
Step 1: S = {1, 2, 3, 4, 5, 6}. Total = 6.
Step 2: Even = {2, 4, 6}, count 3. P(even) = 3/6 = 1/2.
Step 3: Less than 3 = {1, 2}, count 2. P(<3) = 2/6 = 1/3.
Step 4: P(6) = 1/6. P(not 6) = 1 − 1/6 = 5/6.
Check: each value sits between 0 and 1.
3.1
P(heads) = 1/2 = 0.5.
3.2
P(heart) = 13/52 = 1/4 = 0.25.
3.3
Odd = {1, 3, 5, 7}, count 4. P(odd) = 4/8 = 1/2 = 0.5.
3.4
Vowels = {A, E}, count 2. P(vowel) = 2/6 = 1/3.
3.5 — Coin flipped twice
S = {HH, HT, TH, TT}, total = 4.
(a) P(HH) = 1/4.
(b) "At least one tail" = {HT, TH, TT}, count 3. P = 3/4. (Alternative: 1 − P(HH) = 1 − 1/4 = 3/4.)
3.6 — Complement
P(no rain) = 1 − 0.3 = 0.7 = 70 %.
3.7 — Two-flip sample space (the trap)
The mistake is treating HT and TH as the SAME outcome. The order matters: HT means "first heads then tails", TH means "first tails then heads" — these are different sequences. Correct sample space = {HH, HT, TH, TT}, four outcomes. P(exactly one head) = 2/4 = 1/2.
3.8 — Probability cannot exceed 1
Probability is defined as "a measure of how likely an event is to occur, from 0 (impossible) to 1 (certain)". The maximum any probability can take is 1. P = 1.5 is meaningless because there is no event "more likely than certain".