# Can we use Venn diagrams to 'prove' theorems regarding events?

Consider the equality $P(A \cup B) = P(A)+P(B)-P(A \cap B)$.

We can prove this in a number of ways (from the probability axioms, measure theoretically etc.)

However a simple Venn diagram gives an easy geometric explanation.

My question is, can a Venn diagram be used to establish a formal proof of a probability equation, such as the addition rule?

• What concept of "formal proof" do you have in mind? – whuber Feb 19 '18 at 15:20

## 1 Answer

A Venn diagram can help guide our thoughts and intuition. But any Venn diagram shows just one composition of events (sets) $A, B \subseteq \Omega$. For example, the figure below

shows:

1. $A, B$ as circles with equal radius and area (I chose them this way),

2. $A, B$ having non-empty intersection.

But the addition rule,

$$P(A \cup B) = P(A) + P(B) - P(A \cap B),$$

holds generally for any $A,B \subseteq \Omega$, including but not limited to:

• $P(A) \ll P(B)$,
• $A \cap B = \emptyset$,
• $A = \emptyset$,
• $B = A^c$,
• etc.

To cover all these cases in one sweep (proof), we do not rely on Venn diagrams but on the established laws of Logic.

• In the conventional meanings of "realization" and "event" it makes no sense to write that "any Venn diagram shows just one realization of events," leading one to wonder what claim you are trying to make. In fact, this Venn diagram is a schematic that depicts the most general possible set-theoretic relationship between two events, understanding that any (even all) of the four regions in the diagram may be empty. It does not directly convey any information about probability. – whuber Feb 21 '18 at 18:33
• Thanks for your thoughts. 1) Edited for clarity: "realization" replaced by "composition". 2) The point that I am making is that the addition rule holds generally for all compositions of $A,B$ not just "the most general" one (whatever that may be). – Jim Feb 22 '18 at 22:13
• Could you explain what you mean by "composition"? That's not a standard term in set theory (except in its meaning of composition of functions, which wouldn't apply here). – whuber Feb 23 '18 at 15:05
• @whuber you may read it as "choice". – Jim Feb 23 '18 at 16:04
• That reading would, unfortunately, render your answer incorrect, at least not without further elaboration of what you mean: as far as I can see, a Venn diagram does not appear to show just one "choice" of events. It abstractly represents any two events. – whuber Feb 23 '18 at 16:08