Venn, table and tree: three perspectives

When we study two events $A$ and $B$, we have three different graphical tools at our disposal. They are not three separate topics: they are three ways of looking at the same mathematical object.

Observation — One and the same mathematical object, three perspectives

A two-event probabilistic situation $A,B$ is a single mathematical object — the four joint probabilities $P(A\cap B),\ P(A\cap\overline{B}),\ P(\overline{A}\cap B),\ P(\overline{A}\cap\overline{B})$, which sum to $1$ — that we can view from three distinct perspectives:

• Venn diagram: visual, symmetric between $A$ and $B$. The four zones are directly the joint probabilities. It is the most intuitive representation for grasping the overlap structure at a glance.
• Contingency table: compact, saves space. The same numbers as the Venn diagram, but in $2\times 2$ matrix form with the marginals on the rows and the columns. It is the most convenient one for explicit calculations on paper.
• Tree diagram: asymmetric, conditional, sequential. On the branches there always appear conditional probabilities: first level $=P(A_i)$ marginals, second level $=P(B_j\mid A_i)$. It is the most natural representation for sequential experiments (successive throws, draws, diagnostic tests).

The three representations encode the same information and can be converted into one another. Learning the steps is the operational key to tackling almost any exercise.

Links

Topics: Probability
Concepts: Tree diagram · Venn diagram · Joint probability · Contingency table
Methods: Moving between representations
Skills: Probability calculation · Interpreting a graph