Probability from tables and conditional groups
Identify the sample space, restrict it when a condition applies, and recover missing frequencies from probabilities.
What you will tackle
- Interpret relative frequency and probabilities
- Use one- and two-way tables and area models
- Find conditional probabilities with the correct denominator
- Infer missing counts from probabilities
The denominator follows the selection
Suppose a two-way table has 12 biology students who play sport, 8 biology students who do not, 6 other students who play sport and 14 others who do not. There are 40 students total. The probability of sport is 18/40; conditional on biology, it is 12/20. An area model can show the same intersections as rectangles whose areas represent counts or proportions.
Use totals to recover missing entries
If 30% of a group of 50 meets a condition, the count is 15. If 6 of the 15 also meets another condition, that conditional probability is 6/15. Probabilities of disjoint outcomes can be added; overlapping events need their intersection subtracted to avoid double counting.
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
Using counts 12,8,6,14 in the order given above, find P(biology), P(sport), P(sport | biology), and P(biology or sport).
Show a starting hint
Draw the four table cells and their marginal totals.
Compare with the explained answer
- Biology total is 20 of 40; sport total is 18 of 40.
- The conditioned biology group has 12 sports participants out of 20.
- The union count is 20 + 18 − 12 = 26.
Answer / response criteria: 1/2; 9/20; 3/5; 13/20.
Independent practice
Independent 1
In a class of 60, 40% study music. Nine music students also study art. Find the number studying music and P(art | music).
Compare with the explained answer
- The music count is 0.4 × 60 = 24.
- The condition restricts the denominator to 24.
- 9/24 simplifies to 3/8.
Answer / response criteria: 24 students; 3/8.
Application practice
Application 1
A probability model assigns 1/4 of a 200 m² display to category A and 3/5 of A to a patterned subregion. Find both areas and the probability that a uniformly chosen point lies in the patterned subregion.
Compare with the explained answer
- A occupies 200/4 = 50 m².
- The patterned part is (3/5)·50 = 30 m².
- Its share of the whole is 30/200 = 3/20; uniform selection is required for area to represent probability.
Answer / response criteria: A: 50 m²; patterned: 30 m²; probability 3/20.
What can you explain now?
A probability model assigns 1/4 of a 200 m² display to category A and 3/5 of A to a patterned subregion. Find both areas and the probability that a uniformly chosen point lies in the patterned subregion.
Compare with the explanation
A: 50 m²; patterned: 30 m²; probability 3/20.
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