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.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide