The six conversions

Once an initial representation has been chosen, the other two are obtained through elementary calculations. Let us summarise the six possible transformations among table, Venn and tree.

In brief — Six steps between the representations

Let us denote by $n$ the total (if we work in absolute numbers) or $1$ (if we work in probabilities). The four joint probabilities are called $p_{ij}=P(A_i\cap B_j)$ with $i,j\in\{1,2\}$.

• Table $\to$ Venn: copy the four joint cells into the four zones of the Venn diagram. The total in the background is $\Omega=1$.
• Table $\to$ Tree: for each row of the table, compute the conditional frequencies. For example, the branch $A_1\to B_1$ has weight $P(B_1\mid A_1)=p_{11}/(p_{11}+p_{12})$ (cell over row total). First level: marginals $P(A_i)=$ row total $/\,n$.
• Venn $\to$ Table: transcribe the zones into the cells; the marginals are the row and column sums.
• Venn $\to$ Tree: for each zone, compute $P(B\mid A)=|A\cap B|/|A|$ (intersection zone over the zone of the conditioning event).
• Tree $\to$ Table: reconstruct the joint probabilities as a product along the branches $p_{ij}=P(A_i)\cdot P(B_j\mid A_i)$, then marginalise by summing rows and columns.
• Tree $\to$ Venn: the same thing, but write the branch products directly into the four zones.

To these six should be added the seventh conversion, already seen when studying the inversion of the tree: tree $\to$ inverted tree. It changes the first level of conditioning — from $\{A_i\}$ to $\{B_j\}$ — and is Bayes’ theorem read as a geometric operation.

Observation — Which representation to choose?

The table is the most compact: use it when you have little space or have to do quick calculations. The Venn diagram is the most visual: use it when you need an immediate intuition of the overlap. The tree is the most sequential: use it when the problem is formulated as a staged experiment (“draw a ball, then another”) or when you are given the conditionals $P(B\mid A_i)$ and the marginals $P(A_i)$ directly.

Links

Topics: Probability
Concepts: Tree diagram · Venn diagram · Moving between representations · Contingency table
Methods: Moving between representations
Skills: Probability calculation · Modelling