Complements, Venn counts and independence
Count overlapping groups once, use a complement, and test whether one event changes the chance of another.
What you will tackle
- Read and build Venn-region counts
- Use complements and inclusion–exclusion
- Define independence using conditional or joint probability
- Distinguish independence from mutually exclusive events
Disjoint regions prevent double counting
A two-set Venn diagram separates A only, both, B only and neither. The union count is n(A) + n(B) − n(A ∩ B). A complement contains outcomes outside the event, so P(not A) = 1 − P(A) when the whole sample space has probability one.
Independence is a numerical condition
Events are independent when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A) when P(B) > 0. Mutually exclusive events have no shared outcomes. Two positive-probability mutually exclusive events are not independent: learning one occurred makes the other impossible.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
In the displayed 40-person Venn example, 20 study music, 18 art and 8 both. Find music only, art only, union and neither. Test whether membership in the two groups is independent.
Show a starting hint
Keep the overlap as a separate region.
Compare with the explained answer
- Music only is 20 − 8 = 12; art only is 18 − 8 = 10.
- The union is 20 + 18 − 8 = 30; neither is 40 − 30 = 10.
- P(both) = 8/40 = 0.2, while P(music)P(art) = (20/40)(18/40) = 0.225. The unequal values show dependence.
Answer / response criteria: 12,10,30,10; not independent.
Independent practice
Independent 1
A fair die is rolled. A means an even result and B means a result above 4. Find P(not A), P(A ∩ B) and P(A|B), and decide independence.
Compare with the explained answer
- P(not A) = 3/6 = 1/2.
- The intersection is {6}, so its probability is 1/6; B = {5,6}, giving P(A|B) = 1/2.
- P(A)P(B) = (1/2)(1/3) = 1/6, so the events are independent despite sharing an outcome.
Answer / response criteria: 1/2; 1/6; 1/2; independent.
Application practice
Application 1
A 100-person survey has 60 bus users, 50 bicycle users and 30 using both. Build the four Venn counts, find the chance of neither and the chance of cycling given bus use, then test independence. Can these data establish that bus use causes cycling?
Compare with the explained answer
- Bus only 30, both 30, bicycle only 20 and neither 20; P(neither) = 0.2.
- P(cycle|bus) = 30/60 = 0.5, equal to P(cycle) = 0.5, so these sample events satisfy independence.
- This arithmetic describes the sample; an observational survey does not establish a causal mechanism.
Answer / response criteria: 30,30,20,20; 0.2; 0.5; independent within the given sample; no causal conclusion.
What can you explain now?
A 100-person survey has 60 bus users, 50 bicycle users and 30 using both. Build the four Venn counts, find the chance of neither and the chance of cycling given bus use, then test independence. Can these data establish that bus use causes cycling?
Compare with the explanation
30,30,20,20; 0.2; 0.5; independent within the given sample; no causal conclusion.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide