Conditional probability: Venn diagrams and two-way tables
Conditional probability is a higher-tier topic on OCR J560 Papers 2 and 3. It asks "given that event A has happened, what is the probability of event B?" The Venn diagram and two-way table are the most common OCR representations.
Conditional probability notation
P(B | A) means "the probability of B given that A has occurred."
Formula: P(B | A) = P(A and B) / PA
Venn diagrams
A Venn diagram uses overlapping circles inside a rectangle (the universal set ε).
Key regions:
- Inside A only (not B): A∩B' (A-not-B)
- Inside B only (not A): A'∩B
- Inside both (the overlap): A∩B
- Outside both: A'∩B' (neither)
All values in a Venn diagram sum to the total sample size.
Calculating probabilities from Venn diagrams:
PA = (all in circle A) / total. P(A and B) = (overlap region) / total. P(A or B) = (everything in either circle) / total. P(A | B) = (overlap) / (all of B).
Example: 50 students; 30 study French (F), 20 study Spanish (S), 8 study both.
In F only: 30 − 8 = 22. In S only: 20 − 8 = 12. In both: 8. In neither: 50 − 22 − 8 − 12 = 8.
P(F | S) = P(F and S)/P(S) = (8/50)/(20/50) = 8/20 = 2/5.
Two-way tables
Present frequencies or probabilities for two categorical variables.
| Likes sport | Does not like sport | Total | |
|---|---|---|---|
| Male | 15 | 10 | 25 |
| Female | 8 | 17 | 25 |
| Total | 23 | 27 | 50 |
P(male | likes sport) = 15/23 (restricting to the "likes sport" column, find males).
P(likes sport | female) = 8/25 (restricting to females, find those who like sport).
Mutually exclusive and exhaustive events
- Mutually exclusive: A and B cannot both happen → P(A and B) = 0; P(A or B) = PA + PB.
- Exhaustive: A and B cover all possibilities → PA + PB = 1.
Common OCR exam mistakes
- Confusing P(A | B) and P(B | A): order matters! P(F | S) ≠ P(S | F).
- Using the wrong denominator in conditional probability — restrict to the GIVEN condition.
- Venn diagram values not summing to total — always check.
- Confusing "or" (union) with "and" (intersection) in probability calculations.
- Double-counting the overlap in Venn diagrams: P(A or B) ≠ PA + PB when A and B overlap.
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