The Relationship of Venn Diagrams to Tree Diagrams--Independent Events
Any probability problem that can be represented by a Venn diagram can also be represented by a tree diagram and vice versa. The diagrams at right are set up for independent events, but will look very similar for independent events.
Note that specific leaves on the tree correspond to specific areas of the Venn diagram. This is true for both independent and conditional events.
For Independent Events:
The relationships below show how the tree diagram for 2 independent events is related to the diagram for 2 conditional events. Note that for independent events, the second set of branches are identical regardless of whether event A is or isn't true. This is not the case for conditional events.
P(B) = P(B|A) = P(B|AC)
- P(A∩B) = P(A) • P(B)
- P(AUB) = P(A) + P(B) - [P(A) • P(B)]
For conditional events, non of the above relationships will be met.
Example:
- P(A) = 49 %, probability of being male
- P(B) = 20 %, probability of blue eyes
- P(A∩B) = 9.8 %, probability of being a male and having blue eyes
- P(AUB) = 59.2 %, probability of being a male or having blue eyes
