E8.4 Conditional probability
- Syllabus
- 0580–2028–2029
- Topic
- E8.4
- Level
- Extended
Learning objectives
Calculate probability after a condition is known
A condition tells you that only part of the original sample space is still possible. Ignore every outcome outside that condition, then compare the required outcomes with the new, smaller total.
First identify the outcomes that satisfy the given condition. Use all of them as the new denominator. Within that restricted group, count or weight the outcomes that also satisfy the target event; these form the numerator.
| Representation | Restricted denominator | Required numerator |
|---|---|---|
| Venn diagram | every region inside the set named by the condition | the part of those regions also satisfying the target |
| table or sample space | the relevant row, column or selected cells | target cells within that restricted total |
| tree diagram | total probability of every complete path consistent with the condition | total probability of the consistent path or paths that also meet the target |
One number is selected from {1,2,4} and one from {3,4,6}, with all nine ordered pairs equally likely. Given that their sum is odd, only (1,4),(1,6),(2,3),(4,3) remain. Exactly one has first number 4, so the required probability is 41.
A Venn diagram shows 18 students in set F and 10 in the overlap of F and T. If the chosen student is known to be in F, the sample space contains only those 18 students. Ten also belong to T, so the probability is 1810=95.
A tree shows 60% of items follow route A and 40% route B. The faulty rates are 5% on A and 10% on B. Faulty path weights are 0.6×0.05=0.03 and 0.4×0.10=0.04. Given that an item is faulty, only these paths remain, so the probability it followed B is 0.03+0.040.04=74.
The condition changes the denominator, not just the numerator. This syllabus requires calculation from Venn diagrams, trees and tables, but does not require special conditional-probability notation or formulas.