C8.3 Probability of combined events
- Syllabus
- 0580–2028–2029
- Topic
- C8.3
- Level
- Core
Learning objectives
Calculate probabilities of combined events
A combined event joins two or more conditions. Choose a representation that lists every possible outcome once, then combine only the outcomes that satisfy the event.
| Representation | Use it when | How probability is found |
|---|---|---|
| sample space diagram | two stages can be arranged as rows and columns | mark the favourable cells, then divide by all equally likely cells |
| two-set Venn diagram | outcomes are classified by membership of A and B | add the required region counts once, then divide by the universal total |
| tree diagram | events happen in stages | multiply probabilities along each required path; add the probabilities of alternative paths |
P(\text{one path})=P(\text{first branch})\times P(\text{second branch}),\qquad P(\text{alternative paths})=\text{sum of their path probabilities}
A bag contains 6 red, 3 green and 1 blue marble. Two marbles are selected with replacement. The probability of green stays 103 on the second stage, so P(green then green)=103×103=1009. Write outcomes at the ends of branches and probabilities beside the branches.
For two fair six-sided dice, a 6×6 sample space has 36 equally likely ordered pairs. A total of 5 occurs at (1,4),(2,3),(3,2),(4,1), so its probability is 364=91. Ordered pairs such as (1,4) and (4,1) are different outcomes.
In a Venn diagram, A or B includes both single-set regions and the overlap, but the overlap is counted once. A and B is only the overlap. This syllabus uses at most two sets, and combined-event tree problems here use replacement, so branch probabilities do not change after the first selection.