Read conditional probabilities from two-way tables
Build disjoint cells from overlapping totals and choose the conditioning row or column as the denominator.
Section 5.3 · Probability
Exam scope and assessment notes
Probability & Statistics 1 (Paper 5, AS & A Level) · Mathematics 9709, 2026–2027, section 5.3. Define the sample space and state its probability assumptions before counting. Fair independent dice give equiprobable ordered pairs; their sums are not equiprobable. Random selection means each specified individual is equally likely under the stated sampling model. Without replacement, later branch probabilities change; with replacement and independent mixing, they remain the same. A′ is a complement, A ∩ B means both, A ∪ B means either or both, and P(A|B) means A given B with P(B) > 0. Mutually exclusive and independent are different properties. The explicit general addition formula is not required by this section; explain overlap through disjoint regions. Conditional probabilities may use P(A|B) = P(A ∩ B)/P(B). Numerical models describe the stipulated experiment, not automatically a real population. Written practice is self-checked; human subject-teacher review remains pending.
Before you begin
- Use intersections and complements.
- Identify the denominator in a conditional probability.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A totals 48 and both totals 30. How many are A-only?
What you will learn
- Complete a two-way count table from event totals.
- Calculate conditional probabilities and distinguish sample descriptions from population claims.
Overlapping totals must be separated into cells
Among 120 recorded students, 48 are in club A, 60 in club B and 30 in both. The disjoint cells are both 30, A-only 18, B-only 30 and neither 42. A row/column table has totals A 48, not-A 72, B 60 and not-B 60. Check all cells add to 120 before taking ratios.
For a student chosen uniformly from this recorded group, P(A|B) = 30/60 = 1/2. Conversely P(B|A) = 30/48 = 5/8. The same intersection is divided by different given-event totals. P(not-A|B) = 30/60 = 1/2 complements P(A|B) within the B group, not within the entire population.
Condition within the correct column or row
In an eighty-person recorded group, both has 24, A-only 16, B-only 8 and neither 32. The B column totals 32, so P(A|B) = 24/32 = 3/4. The not-B column totals 48, so P(A|not-B) = 16/48 = 1/3. Do not use total A = 40 as the denominator for either of these questions.
Uniform selection from a recorded table supports exact probabilities for that finite group. Using observed proportions to predict another population requires a representative sampling model. A voluntary poll of 40 respondents with 18 bus travellers describes a bus proportion 18/40 = 0.45 in that respondent group; it does not automatically establish the same proportion for all students.
Pause and explain
120 students: 48 in A, 60 in B, 30 in both. What is P(A|B)?
Worked example
Among 120 students, 48 belong to A, 60 to B and 30 to both. Choose a student uniformly. Find P(A|B) and P(B|A).
Show the worked solution
- The shared cell has 30 students.
- Conditioning on B gives denominator 60, so P(A|B) = 1/2.
- Conditioning on A gives denominator 48, so P(B|A) = 5/8.
Answer P(A|B) = 1/2; P(B|A) = 5/8.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Among 120 students, 48 belong to A, 60 to B and 30 to both. Choose uniformly from the group. Find P(A|B) and P(B|A).
Give me a hint
Use the shared count with the appropriate club total each time.
Compare my reasoning
- The shared numerator is 30 in both calculations.
- P(A|B) = 30/60 = 1/2.
- P(B|A) = 30/48 = 5/8.
1/2 and 5/8 respectively.
Look for these in your work
- Read the condition direction.
- Explain the different denominators.
A table of 80 people has both 24, A-only 16, B-only 8 and neither 32. Choose uniformly. Find P(A|B) and P(A|not-B).
Give me a hint
Add the cells in each conditioning column before calculating its ratio.
Compare my reasoning
- The B total is 24 + 8 = 32, so P(A|B) = 24/32 = 3/4.
- The not-B total is 16 + 32 = 48, so P(A|not-B) = 16/48 = 1/3.
- The different denominators correspond to different information being given.
3/4 and 1/3 respectively.
Look for these in your work
- Complete the conditioning totals.
- Keep the A-only cell distinct from all A members.
A voluntary travel poll has 40 respondents: 18 use a bus, 12 walk and 10 cycle, with one recorded mode per person. Find the bus probability for a uniformly chosen respondent and assess a claim that it is exactly the same for every student population.
Give me a hint
A uniform choice from respondents is different from representative sampling of all students.
Compare my reasoning
- The respondent categories sum to 40.
- A uniformly chosen respondent uses the bus with probability 18/40 = 0.45.
- Voluntary participation can be unrepresentative, so the population claim is unsupported by these counts alone.
0.45 for the recorded respondent group; the general population claim is unsupported.
Look for these in your work
- State which finite group the probability describes.
- Identify the missing representative-sampling assumption.
Common mistakes
- Marginal totals include the shared cell.
- Use the given row or column total, not the whole table.
- A recorded-group proportion does not establish a representative population probability.
What can you explain now?
Why does P(A|B) + P(A′|B) equal one?
Compare with the explanation
The A and not-A cells partition the B conditioning group.
Both fractions use the same positive B total as denominator.
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.
Make it stick
Complete the disjoint cells, check totals, and divide by the given row or column rather than an unrelated marginal.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.