Mathematics • Year 7 • Unit 4 • Lesson 14
Introduction to Probability
Build fluency with the probability formula: P(event) = favourable outcomes ÷ total outcomes. List the sample space, count carefully, and use the complement rule P(not A) = 1 − P(A).
1. I do — fully worked example
Read every line. Each step shows the question to ask and the reason for the answer.
Problem. A standard die is rolled. Find P(rolling a 4).
Step 1 — List the sample space.
A die has faces 1, 2, 3, 4, 5, 6. Sample space S = {1, 2, 3, 4, 5, 6}.
Reason: every outcome must appear exactly once; all equally likely.
Step 2 — Count favourable outcomes.
Event = "roll a 4". Only one outcome matches: {4}. Favourable = 1.
Reason: a favourable outcome is one that makes the event TRUE.
Step 3 — Apply the formula.
P(4) = favourable ÷ total = 1 ÷ 6 = 1/6 ≈ 0.167 ≈ 16.7%.
Reason: probability formula. Check the answer is between 0 and 1.
Answer: P(4) = 1/6 ≈ 0.167.
2. We do — fill in the missing steps
A bag contains 3 red, 5 blue and 2 green marbles. A marble is drawn at random. Find P(red), P(green) and P(not red). Fill in each blank. 5 marks
Step 1 — Total marbles in the bag:
Total = 3 + 5 + 2 = _______ marbles.
Step 2 — P(red):
Favourable (red) = _______. P(red) = ___ / ___ = ___ (simplified) = ___ (decimal).
Step 3 — P(green):
Favourable (green) = _______. P(green) = ___ / ___ = ___ (simplified) = ___ (decimal).
Step 4 — P(not red) using the complement rule:
P(not red) = 1 − P(red) = 1 − ___ = ___ .
Step 5 — Check:
Count blue + green = 5 + 2 = ___ . P(not red) = ___ / 10 . Match? (YES / NO)
3. You do — independent practice
Write probabilities as simplified fractions AND as decimals (to 2 d.p. where needed). Check each answer is between 0 and 1.
Foundation — quick probabilities
3.1 A coin is flipped. What is P(heads)? 1 mark
3.2 A die is rolled. What is P(rolling an even number)? 1 mark
3.3 A bag has 4 red and 6 blue counters. What is P(red)? 1 mark
3.4 A standard pack of 52 cards is shuffled. What is P(drawing a red card)? 1 mark
Standard — list sample spaces, then calculate
3.5 A letter is chosen at random from {A, B, C, D, E}. (i) Write the sample space. (ii) Find P(vowel). 2 marks
3.6 An 8-section spinner is numbered 1–8. (i) Find P(spinning a number greater than 5). (ii) Find P(spinning a prime number) — recall: 2, 3, 5, 7 are prime. 2 marks
Extension — complement rule and checks
3.7 If P(A) = 2/3, find P(not A). Then explain in one sentence what "P(A) + P(not A) = 1" means in words. 2 marks
3.8 A student calculates P(winning) = 1.2 for a game. Explain (i) why this answer must be wrong, and (ii) one likely mistake the student made. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — Marble bag (We do)
Step 1: Total = 3 + 5 + 2 = 10 marbles.
Step 2: Favourable (red) = 3. P(red) = 3/10 = 0.3.
Step 3: Favourable (green) = 2. P(green) = 2/10 = 1/5 = 0.2.
Step 4: P(not red) = 1 − 3/10 = 7/10 = 0.7.
Step 5: Blue + green = 7. P(not red) = 7/10. Match? YES.
3.1 — P(heads)
Sample space {H, T}. P(heads) = 1/2 = 0.5.
3.2 — P(even on a die)
Even faces {2, 4, 6} = 3 favourable. P(even) = 3/6 = 1/2 = 0.5.
3.3 — P(red counter)
Total = 4 + 6 = 10. P(red) = 4/10 = 2/5 = 0.4.
3.4 — P(red card)
26 red cards out of 52. P(red) = 26/52 = 1/2 = 0.5.
3.5 — Vowels from {A, B, C, D, E}
(i) Sample space = {A, B, C, D, E}, 5 letters total.
(ii) Vowels: A, E → 2 favourable. P(vowel) = 2/5 = 0.4.
3.6 — 8-section spinner
(i) Numbers > 5: {6, 7, 8} = 3 favourable. P = 3/8 = 0.375.
(ii) Primes in 1–8: {2, 3, 5, 7} = 4 favourable. P(prime) = 4/8 = 1/2 = 0.5.
3.7 — Complement
P(not A) = 1 − 2/3 = 1/3. In words: every event and its complement together cover all possible outcomes, so their probabilities must sum to 1 (i.e. 100%).
3.8 — P = 1.2 is impossible
(i) Probability must lie between 0 and 1 inclusive — P = 1.2 is greater than 1, which is impossible.
(ii) Likely mistake: the student counted more favourable outcomes than total outcomes (perhaps double-counting some outcomes), or they used the wrong denominator (smaller than the total).
Marking: 1 for "must be between 0 and 1"; 1 for naming impossibility; 1 for a plausible counting error.