Mathematics • Year 7 • Unit 4 • Lesson 20
Data and Chance Synthesis
Build fluency joining data and probability: (i) the PPDAC cycle for a real investigation, (ii) relative frequency = experimental probability, and (iii) expected frequency E = P × n for making predictions from data.
1. I do — fully worked example
Read every line. One problem, three connected ideas.
Problem. A survey of 150 students records their favourite genre: Action 45, Comedy 60, Drama 30, Documentary 15. Find the experimental P(Comedy), then predict the number of "Comedy" picks if the school surveys 600 students.
Step 1 — Confirm the total trials.
Total = 45 + 60 + 30 + 15 = 150 ✓
Reason: cross-check the sum equals the stated sample size.
Step 2 — Compute the relative frequency (= experimental probability).
P(Comedy) = 60 ÷ 150 = 2/5 = 0.4
Reason: relative frequency for a category IS the experimental probability for that category.
Step 3 — Expected frequency for a larger group.
E = P × n = 0.4 × 600 = 240 students
Reason: if the 150 sample is representative, the predicted proportion of "Comedy" picks in 600 is also about 0.4.
Answer: P(Comedy) = 0.4, expected Comedy picks in 600 ≈ 240 students.
2. We do — fill in the missing steps
A bag of 8 marbles contains 3 red. A marble is drawn at random with replacement and noted. The bag is sampled 240 times. Find the expected number of red picks. Fill in each blank. 4 marks
Step 1 — Find the theoretical probability of red.
P(red) = ___ ÷ ___ = _______ (as a decimal)
Step 2 — Identify n (the number of trials).
n = _______
Step 3 — Apply the expected-frequency formula.
E = P × n = ___ × ___ = _______
Step 4 — Sense-check.
The expected count is sensible because it is about ___ of the total, which matches P(red) ≈ ___.
3. You do — independent practice
Show formula lines: relative frequency = freq ÷ total; expected frequency E = P × n.
Foundation — apply E = P × n
3.1 A fair die is rolled 300 times. How many times would you expect to roll a 6? 1 mark
3.2 A team has P(win) = 0.4 per game. In a season of 50 games, how many wins are expected? 1 mark
3.3 A spinner has P(yellow) = 1/4. The spinner is spun 80 times. How many yellow spins are expected? 1 mark
Standard — relative frequency from data
3.4 In a survey of 200 students, 80 prefer Maths over Science. Find the experimental probability that a random student prefers Maths. 1 mark
3.5 A frequency table shows transport to school: bus 90, walk 60, car 36, bike 14 (n = 200). (i) Find P(car). (ii) If the whole Year 7 cohort has 500 students, estimate how many travel by car. 2 marks
3.6 List the five stages of the PPDAC cycle in order, with one short sentence describing each. 2 marks
Extension — push your thinking
3.7 The school cafeteria estimates that 35% of all student orders include a vegetarian option. On a day with 480 orders, (i) find the expected number of vegetarian orders, (ii) explain in one sentence why the actual count might be 158 instead of the expected value. 3 marks
3.8 A bag has 5 red, 2 blue and 3 green marbles. A marble is drawn at random with replacement 200 times. Find the expected counts for red, blue, and green. Verify the three expected counts add to 200. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — Bag of 8 (We do)
Step 1: P(red) = 3 ÷ 8 = 0.375. Step 2: n = 240. Step 3: E = 0.375 × 240 = 90 red picks. Step 4: 90 is about 3/8 of 240, which matches P(red) ≈ 0.375. ✓
3.1 — Die rolled 300 times, expected 6s
E = (1/6) × 300 = 50 sixes.
3.2 — Team wins
E = 0.4 × 50 = 20 wins.
3.3 — Yellow spins
E = (1/4) × 80 = 20 yellow spins.
3.4 — Prefer Maths
P(Maths) = 80 ÷ 200 = 2/5 = 0.4.
3.5 — Transport to school
(i) P(car) = 36 ÷ 200 = 9/50 = 0.18.
(ii) Expected car users in 500 = 0.18 × 500 = 90 students.
3.6 — PPDAC cycle
Problem — state the question you want to answer.
Plan — design how data will be collected (sample size, method).
Data — collect and organise the observations.
Analysis — calculate statistics, draw graphs, look for patterns.
Conclusion — answer the original question using the evidence.
3.7 — Cafeteria vegetarian orders
(i) E = 0.35 × 480 = 168 vegetarian orders.
(ii) The actual count (158) differs because expected frequency is a prediction, not a guarantee — natural variation means real samples fluctuate around the expected value.
3.8 — Bag 5R, 2B, 3G, 200 trials
P(R) = 5/10 = 0.5 → E(R) = 0.5 × 200 = 100.
P(B) = 2/10 = 0.2 → E(B) = 0.2 × 200 = 40.
P(G) = 3/10 = 0.3 → E(G) = 0.3 × 200 = 60.
Sum = 100 + 40 + 60 = 200. ✓