Mathematics • Year 9 • Unit 4 • Lesson 19
Data and Probability — Mixed Challenge
Mix measures of centre, addition/multiplication probability rules, complement and bias-spotting. Choose the right tool, find a classmate's mistake, and design your own representative survey.
1. Mixed problems — pick the right tool
Each question uses a different combination. Decide what the question is asking before reaching for a formula. 3 marks each
1.1 Find the five-number summary (min, Q1, median, Q3, max) for: 5, 8, 10, 12, 15, 18, 25.
1.2 Two dice are rolled. Find P(sum is even) using either a sample-space table or by counting (both even, both odd).
1.3 P(A) = 0.4, P(B) = 0.3, P(A ∩ B) = 0.1. Find (a) P(A ∪ B), (b) P(neither A nor B).
1.4 A bag has 5 black and 4 white balls. Two are drawn without replacement. Find (a) P(both black), (b) P(one of each colour, in either order).
1.5 A school surveys "favourite sport" by asking only members of the basketball team. Identify the bias and explain in one sentence why the results would mislead a P.E. teacher buying new equipment.
1.6 A box plot shows min = 20, Q1 = 30, median = 50, Q3 = 65, max = 90, with a small dot marking an outlier at 10. (a) Describe the distribution in one sentence (skew + outlier). (b) Find the IQR. (c) If a new value of 95 is added (no outlier), how would Q3 move (up, down, or stay)?
2. Find the mistake
A student is asked: "P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A ∪ B)." Their working below. Exactly one line contains a mistake. Spot it, explain, then redo. 3 marks
Student's working:
Line 1: Identify rule: addition rule.
Line 2: P(A ∪ B) = P(A) + P(B) + P(A ∩ B)
Line 3: P(A ∪ B) = 0.6 + 0.5 + 0.2 = 1.3
Line 4: So P(A ∪ B) = 1.3.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong (refer to the lesson's probability rules summary AND the fact that all probabilities lie between 0 and 1).
(c) Re-do the working correctly, including the corrected final value.
Stuck? Even before checking the formula, 1.3 can't be a probability. The correct addition rule SUBTRACTS the intersection — see lesson § "Probability Rules Summary".3. Open-ended challenge — design a representative survey
This question has many valid answers. 4 marks
3.1 Your school principal wants to know "what proportion of students at our school of 720 want a longer lunch break?" Design a survey that will give a trustworthy answer.
Your design must include:
(i) The sampling method (random, stratified, convenience, etc.) and a justification using the lesson's data-collection checklist.
(ii) The sample size (n) and a one-sentence justification for why it's adequate.
(iii) The actual survey question, worded to avoid leading language (give the exact words a student would read).
(iv) One specific source of bias you have actively chosen to avoid, and how your design avoids it.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Five-number summary
Sorted: 5, 8, 10, 12, 15, 18, 25 (already). n = 7.
Median = 4th value = 12. Lower half (5, 8, 10) → Q1 = 8. Upper half (15, 18, 25) → Q3 = 18.
Summary: min = 5, Q1 = 8, median = 12, Q3 = 18, max = 25. (IQR = 10.)
1.2 — Two dice, P(sum even)
Sum is even iff both dice are even OR both odd. Each die has 3 even and 3 odd faces.
Both even: 3 × 3 = 9 outcomes. Both odd: 3 × 3 = 9. Total = 18 favourable out of 36.
P(sum even) = 18/36 = 1/2 = 0.5.
1.3 — Addition + complement
(a) P(A ∪ B) = 0.4 + 0.3 − 0.1 = 0.6.
(b) P(neither) = 1 − P(A ∪ B) = 1 − 0.6 = 0.4.
1.4 — Bag with 5 black, 4 white
(a) P(both black) = (5/9) × (4/8) = 20/72 = 5/18 ≈ 0.278.
(b) P(one of each, either order) = P(black then white) + P(white then black) = (5/9)(4/8) + (4/9)(5/8) = 20/72 + 20/72 = 40/72 = 5/9 ≈ 0.556.
1.5 — Basketball-team survey
Selection bias (sometimes called sampling bias). Asking only the basketball team massively over-represents basketball as a favourite. A P.E. teacher buying equipment using this data would buy too many basketballs and not enough soccer balls, cricket gear, etc.
1.6 — Box plot description
(a) Roughly symmetric box with a slight stretch on the low side (median − Q1 = 20, Q3 − median = 15), and a low outlier at 10 — overall slightly left-skewed with one low outlier.
(b) IQR = Q3 − Q1 = 65 − 30 = 35.
(c) Adding 95 (between current max 90 and far above Q3 = 65) shifts the upper portion of the data up; Q3 would increase slightly (or stay roughly the same depending on the data set size).
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The correct addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B), not "+". The student double-counted the overlap. The giveaway is that Line 3 gives 1.3, and probabilities must lie between 0 and 1 — that alone should have triggered a recheck.
(c) Correct working:
P(A ∪ B) = 0.6 + 0.5 − 0.2 = 0.9. (0 ≤ 0.9 ≤ 1. ✓)
3 — Open-ended challenge (sample solution)
(i) Stratified random sampling by year group. The school has 6 year groups; pick proportional numbers from each so all year groups are represented. The lesson's checklist asks "was the sample representative?" and "was the sampling method appropriate?" — stratified sampling answers both.
(ii) n = 100 students (≈ 17 from each year group). For 720 students, 100 is a generous sample — enough to be confident in within ±5% of the true proportion. The lesson's checklist asks "was the sample size adequate?" — 100 is.
(iii) Exact question wording: "Should the school's lunch break be longer than it currently is? □ Yes □ No □ Unsure" — neutral, no leading language, with an "unsure" option to avoid forcing fence-sitters.
(iv) Bias avoided: response bias from leading questions. Avoided by phrasing the question neutrally (not "Would you like a much-needed longer break?", which leads to YES). Also avoided self-selection by giving the survey to the randomly-selected students directly in class, rather than letting students opt in.
Marking: 1 mark per part (i)–(iv). Award full marks for any reasonable design that meets all four conditions.