Sheet № 246 · Foundation + Higher · AQA · Edexcel · OCR
Probability from Two-Way Tables –
Probability from two-way tables is a staple question on GCSE Maths papers at Foundation and Higher tier for AQA, Edexcel and OCR. At Foundation level you calculate straightforward probabilities by reading values from the table. At Higher level, you face "given that" (conditional) probability questions where the denominator changes. This g
§Key definitions
Question:
100 students were surveyed about their favourite subject.
(a)
Maths total = 40. Grand total = 100. P(Maths) = 40/100 = 2/5.
(b)
Girls who prefer English = 20. P(girl and English) = 20/100 = 1/5.
Answer:
(a) 2/5 (b) 1/5.
(c)
Condition: part-time. Part-time total = 70. Part-time Sales = 20. P(Sales | part-time) = 20/70 = 2/7.
§Formulas to memorise
P(A) = frequency of A ÷ grand total
P(A given B) = frequency of (A and B) ÷ total of B
Simple probability — divide the cell frequency by the grand total.
Conditional probability ("given that") — divide the cell frequency by the row or column total that matches the given condition.
Read the table — identify what each row and column represents.
Identify the event — what outcome does the question ask about?
Decide the denominator: — For simple probability, use the grand total.
Find the numerator — the frequency of the specific cell or cells that satisfy the event.
Calculate and simplify — the fraction.
Worked example
100 students were surveyed about their favourite subject. | | Maths | English | Science | Total | |---|---|---|---|---| | Boys | 22 | 12 | 16 | 50 | | Girls | 18 | 20 | 12 | 50 | | Total | 40 | 32 | 28 | 100 | A student is chosen at random.
Working:
⚠ Common mistakes
- ✗Using the grand total as the denominator for conditional probability. When the question says "given that" or restricts to a specific group, the denominator is the row or column total for that group, not the grand total.
- ✗Misidentifying which row or column to use. Read "given that" carefully — the word after "given that" tells you which total to use as the denominator.
- ✗Not simplifying fractions. Always simplify your probability to its lowest terms.
✦ Exam tips
- →The phrase "given that" always signals conditional probability — change your denominator.
- →Phrases like "of the boys" or "among the full-time workers" also signal conditional probability even without using the exact words "given that."
- →At Foundation level, the denominator is almost always the grand total. At Higher level, watch for the conditional twist.
- →For basic two-way table skills, see two-way tables. For more on conditional probability, see conditional probability.