Use a Venn diagram when events are represented as overlapping sets, and use a tree diagram when outcomes occur in stages. In both cases, place probabilities or frequencies correctly before applying the relevant probability rules.
Choosing the Diagram
| Diagram | Best used for | Main calculation rule |
|---|---|---|
| Venn diagram | Combined events such as “A or B”, “A and B”, or “A given B” | Add regions, avoiding double-counting |
| Tree diagram | Successive events, especially with or without replacement | Multiply along branches and add separate paths |
A Venn diagram partitions the sample space into disjoint regions. Fill the intersection first, then the exclusive parts, and finally the region outside both sets; all frequencies must total the sample size.
Using a Venn Diagram
Suppose 40 students are surveyed. Of these, 22 study music, 18 play sport, and 8 do both.
The intersection must be counted only once:
For conditional probability, restrict the denominator to the given event. The probability that a student studies music, given that they play sport, is
Using a Tree Diagram
Suppose a product comes from factory A with probability or factory B with probability . The probability of passing inspection is from A and from B.
At every split, branch probabilities must sum to one. If an item is not replaced, both the numerator and denominator change on the second draw because the trials are dependent.
Multiply probabilities along each path:
Then add the mutually exclusive paths:
A common misconception is that mutually exclusive events are independent. They are not: mutually exclusive events cannot occur together, whereas independent events do not affect each other’s probabilities.
IB Exam Technique
This is SL 4.6: Combined events and conditional probability and can appear on either GDC-active paper. Examiners expect labelled regions or branches, multiplication along tree paths, addition across alternative paths, and an updated second-stage probability when sampling without replacement.