Mathematics Advanced • Year 12 • Module 5 • Lesson 4
Independence and Mutual Exclusivity
Past-paper style: classifying event pairs, plus an extended response critiquing a misuse of "independent" in a newspaper claim.
1. Short-answer questions
1.1 A fair die is rolled. Let A = "roll a multiple of 2" and B = "roll a multiple of 3".
(a) List the outcomes in A, B and A ∩ B and find P(A), P(B) and P(A ∩ B).
(b) Determine, with calculations, whether A and B are independent. 3 marks Band 3
1.2 Events A and B have P(A) = 0.5, P(B) = 0.3 and P(A ∩ B) = 0. Decide whether A and B are (a) mutually exclusive; (b) independent. Justify each answer with one line of working. 2 marks Band 3-4
1.3 Two events with P(A) = 0.4 and P(B) = 0.6 are independent. Find P(A ∩ B), P(A ∪ B) and P(A | B). Comment in one sentence on how P(A | B) relates to P(A) for independent events. 4 marks Band 4
Stuck on 1.3? For independent events, P(A ∩ B) = P(A) × P(B).2. Extended response
2.1 A clinical trial of a new asthma drug records two adverse events: drowsiness (D) and dry mouth (M). Long-run data shows:
- P(D) = 0.30 P(M) = 0.20 P(D ∩ M) = 0.06
A health columnist writes:
"Drowsiness and dry mouth are completely separate side effects — they are mutually exclusive, so they are also independent."
(a) Test whether D and M are independent.
(b) Test whether D and M are mutually exclusive.
(c) Identify every mathematical error in the columnist's statement and explain what each error reveals about the common confusion between independence and mutual exclusivity.
(d) Rewrite the columnist's sentence in one mathematically correct sentence that conveys the actual finding from (a) and (b). 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark — calculates P(D) × P(M) = 0.06 and compares to P(D ∩ M) = 0.06; concludes independent.
Part (b) — 1 mark — observes P(D ∩ M) = 0.06 ≠ 0; concludes not mutually exclusive.
Part (c) — 3 marks
• 1 mark — identifies error 1: claim "they are mutually exclusive" is contradicted by P(D ∩ M) = 0.06 ≠ 0.
• 1 mark — identifies error 2: the implication "mutually exclusive ⇒ independent" is false; with positive probabilities, mutual exclusivity actually forbids independence (it is the maximum dependence).
• 1 mark — explains the underlying confusion: "separate / unrelated" in everyday English ≠ statistical independence; the columnist swaps possibility (mutual exclusivity) for information (independence).
Part (d) — 2 marks — one correct sentence such as: "In this trial, drowsiness and dry mouth can both occur (P(both) = 0.06), and they occur together exactly as often as chance would predict — so they are independent, not mutually exclusive." (1 mark for "can both occur / not mutually exclusive"; 1 mark for explicit "independent" with the joint = product fact.)
Your response:
For (c), look for both a factual error ("mutually exclusive" is false here) and a logical error ("mutually exclusive ⇒ independent" is false in general).How did this worksheet feel?
What I'll revisit before next class:
1.1 — Multiples of 2 and 3 on a die (3 marks)
Sample response. (a) A = {2, 4, 6}, B = {3, 6}, A ∩ B = {6}. P(A) = 3/6 = 1/2; P(B) = 2/6 = 1/3; P(A ∩ B) = 1/6.
(b) Independence test: P(A) × P(B) = (1/2)(1/3) = 1/6 = P(A ∩ B) ✓ → A and B are independent.
Marking notes. 1 mark — lists all three sets correctly. 1 mark — computes the three probabilities. 1 mark — performs the independence test and states the conclusion. Common error: forgetting to check the test and assuming dependence from the intersection alone.
1.2 — Mutually exclusive with positive probabilities (2 marks)
Sample response. (a) P(A ∩ B) = 0 → mutually exclusive. (b) P(A) × P(B) = 0.15 ≠ 0 = P(A ∩ B) → not independent. (Confirms the lesson theorem: mutually exclusive events with positive probability are always dependent.)
Marking notes. 1 mark — correct conclusion for mutual exclusivity. 1 mark — correct conclusion for independence with the comparison shown. Common error: stating "mutually exclusive, so independent".
1.3 — Independent events with given P's (4 marks)
Sample response. Independence gives P(A ∩ B) = P(A) × P(B) = 0.4 × 0.6 = 0.24. P(A ∪ B) = 0.4 + 0.6 − 0.24 = 0.76. P(A | B) = P(A ∩ B)/P(B) = 0.24/0.6 = 0.4 = P(A).
Comment: For independent events, P(A | B) equals P(A) — knowing B occurred gives no information about A.
Marking notes. 1 mark — P(A ∩ B) using independence. 1 mark — P(A ∪ B). 1 mark — P(A | B) = 0.4. 1 mark — explicit comment that P(A | B) = P(A) is the conditional form of independence.
2.1 — Asthma trial side effects (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Independence test. P(D) × P(M) = 0.30 × 0.20 = 0.06 = P(D ∩ M) ✓. Therefore D and M are independent. [1 mark]
(b) Mutual exclusivity test. P(D ∩ M) = 0.06 ≠ 0, so D and M are not mutually exclusive. [1 mark]
(c) Errors in the columnist's claim.
Error 1 (factual): The claim "they are mutually exclusive" is wrong. The data says P(D ∩ M) = 0.06, meaning 6% of patients suffer both side effects, so they can co-occur. [1 mark]
Error 2 (logical): Even if D and M were mutually exclusive (with positive probabilities), the implication "mutually exclusive ⇒ independent" would be false. By the lesson theorem, mutually exclusive events with positive probability are maximally dependent: knowing one occurred guarantees the other did not. [1 mark]
Underlying confusion: The columnist is using "separate / unrelated" in its everyday sense, which sounds like independence (information). But mutual exclusivity is about possibility (can both happen?). These two meanings — information vs possibility — are the lesson's key distinction. [1 mark]
(d) Corrected one-sentence claim. "Drowsiness and dry mouth can both occur in the same patient (about 6% of patients have both), and they occur together exactly as often as chance alone would predict — so the two side effects are statistically independent, not mutually exclusive." [2 marks — both 'can both occur' and 'independent' present with the joint = product fact.]
Total: 7/7.
Band descriptors for marker.
Band 3: Tests (a) and (b) correctly numerically; critiques only one error (typically the factual one). ≈ 3-4 marks.
Band 4: Both errors identified but explanation conflates them; rewrite is technically correct but omits the joint-equals-product fact. ≈ 5-6 marks.
Band 5: Both errors clearly separated; rewrite mentions independence but not the "can both occur" point. ≈ 6 marks.
Band 6: Factual error, logical error and underlying confusion all explicitly named; rewrite captures both "can both occur" and "independent" with the joint=product fact. 7/7.