Guided example — School of 1024 students

Example — School of 1024 students: all three representations

In a school of $1024$ students there are two streams (science $A$, classics $\overline{A}$) and the students may play a sport $B$ or not $\overline{B}$. Data: $P(A)=0,4$ (science), $P(B)=0,5$ (sport), $P(A\cap B)=0,1$ (science and sport). Let us build the three representations step by step.

Step 1 — Venn $\to$ Table. From the Venn diagram we obtain the four zones:

$P(A\cap B)=0,1,\qquad P(A\cap\overline{B})=0,4-0,1=0,3,$

$P(\overline{A}\cap B)=0,5-0,1=0,4,\qquad P(\overline{A}\cap\overline{B})=1-0,8=0,2.$

We transfer them into the cells of the table (absolute numbers $=$ probability $\times 1024$):

	$A$ (science)	$\overline{A}$ (classics)	Total
$B$ (sport)	$0,1\cdot 1024 = 102$	$0,4\cdot 1024 = 410$	$512$
$\overline{B}$ (no sport)	$0,3\cdot 1024 = 307$	$0,2\cdot 1024 = 205$	$512$
Total	$410$	$614$	$1024$

(The numbers are rounded to the nearest integer: $1024$ is not divisible by $10$.) The marginals are the row and column sums: $P(A)=410/1024\approx 0,4$, $P(B)=512/1024=0,5$.

Step 2 — Table $\to$ Tree. To obtain the tree (conditioned on $A$) we compute the conditional frequencies “cell over the total of the marginal of $A$”. From the table:

$P(B\mid A) = \frac{P(A\cap B)}{P(A)} = \frac{0,1}{0,4} = \tfrac{1}{4},\qquad P(\overline{B}\mid A) = \tfrac{3}{4},$

$P(B\mid \overline{A}) = \frac{0,4}{0,6} = \tfrac{2}{3},\qquad P(\overline{B}\mid \overline{A}) = \tfrac{1}{3}.$

The tree turns out to be:

The tree conditioned on $A$: on the branches the conditional probabilities, on the leaves in red the branch products (the joint probabilities). Check: $0,1+0,3+0,4+0,2=1$ ✓.

Conclusion. The three pictures — Venn (four zones), table ($2\times 2$ cells), tree (four leaves) — carry exactly the same information: $0,1$, $0,3$, $0,4$, $0,2$. Only the perspective changes: absolute (Venn/table) versus conditional (tree).

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