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. ✓