1.6 Venn Mnemonic for the Rules of Probability

Venn-style diagram: a shaded box labeled Universe contains a single rectangle A occupying 1/4 of its area; the region outside the rectangle is labeled 'not A'. — **Figure 1.2.** Venn diagram of a statement, $A$ , in a *Universe* of all possible statements. It is customary to think of the area of the *Universe* to be equal to 1 so that we can treat the actual areas as fractional areas representing the probability of statements like $P (A)$ . In this image, $A$ takes up 1/4 of the *Universe*, so that $P open parenthesis A close parenthesis equals 1 / 4$ . Also shown is the negation rule. $P (A) + P (not A) = 1$ or “inside” of $A$ + “outside” of $A$ adds up to everything.

It is often useful to have a picture to represent the mathematics, so that it is easier to remember the equations and to understand their meaning. It is common to use what is called a Venn Diagram to represent probabilities in an intuitive, graphical way. The idea is that probabilities are represented as the fractional area of simple geometric shapes. We can then find a picture representation of each of the rules of probability. We start by looking at a sample Venn Diagram, in Figure 1.2.

Venn-style diagram of two overlapping rectangles in a Universe box: A covers 1/4 and B covers 1/8 of the area, their overlap is 1/16 and is labeled 'A and B'; a brace spanning both rectangles is labeled 'A or B'. — **Figure 1.3.** Venn diagram of the sum and product. The rectangle $B$ takes up 1/8 of the *Universe*, and the rectangle $A$ takes up 1/4 of the *Universe*. Their overlap here is 1/16 of the *Universe*, and represents $P (A and B)$ . Their total area of 5/16 of the *Universe* represents $P (A or B)$ .

The fractional area of the rectangle $A$ represents the probability $P (A)$ , and can be thought of as a probability of one of the statements we've explored, such as $P (♥)$ . This diagram is strictly a mnemonic, because the individual points on the diagram are not properly defined. The diagram in Figure 1.2 also represents the Negation Rule (Equation 1.7),

$\begin{matrix} P (A) + P (not A) = 1 \end{matrix}$

In the diagram it is easy to see that the sum of the areas inside of $A$ (i.e. 1/4) and outside of $A$ (i.e. 3/4) cover the entire area of the Universe of statements, and thus add up to 1.

Figure 1.3 shows the diagram which can help us remember the sum and product rules. The Sum Rule (Equation 1.10)

$\begin{matrix} P (A or B) = P (A) + P (B) - P (A and B) \end{matrix}$

is represented in the total area occupied by the rectangles $A$ and $B$ , and makes up all of $A$ (i.e. 1/4) and the half of $B$ sticking out (i.e. 1/8-1/16=1/16) yielding $P open parenthesis A or B close parenthesis equals 5 / 16$ . This is also the area of each added up (1/4+1/8), but subtracting the intersection (1/16) because otherwise it is counted twice. Adding the areas this way directly parallels the Sum Rule.

Conditional probabilities, like those that come into the Product Rule (Equation 1.8) and Bayes Rule (Equation 1.14) are a little more challenging to visualize. In Figure 1.4, $P (A | B)$ is represented by the fraction of the darker area (which was originally part of $A$ ) compared not to the Universe but to the area of $B$ , and thus represents $P open parenthesis A vertical bar B close parenthesis equals 1 / 2$ . In a way, it is as if the conditional symbol, “ $|$ ,” defines the Universe with which to make the comparisons. On the left of Figure 1.4, the same darker area that was originally part of $B$ represents $P (B | A)$ making up 1/4 of the area of $A$ . Thus $P open parenthesis B vertical bar A close parenthesis equals 1 / 4$ . The Product Rule (Equation 1.8) then follows,

$\begin{matrix} P (A and B) = \underset{1 / 2}{\underset{⏟}{P (A | B)}} \underset{1 / 8}{\underset{⏟}{P (B)}} = \underset{1 / 4}{\underset{⏟}{P (B | A)}} \underset{1 / 4}{\underset{⏟}{P (A)}} = \frac{1}{16} \end{matrix}$

We can further see the special case of mutually exclusive statements shown in Figure 1.5. The Sum Rule for Exclusive Events (Equation 1.11) is simply the sum of the two areas because there is no overlap

Two conditional-probability panels. Left: inside rectangle A, the darker strip that A shares with B is labeled B|A and makes up 1/4 of A's area. Right: inside rectangle B, the same shared strip is labeled A|B and makes up 1/2 of B's area. — **Figure 1.4.** Venn diagram of conditional probabilities, $P (A | B)$ and $P (B | A)$ . (Right) $P (A | B)$ is represented by the fraction of the darker area (which was originally part of $A$ ) compared not to the *Universe* but to the area of $B$ , and thus represents $P open parenthesis A vertical bar B close parenthesis equals 1 / 2$ . In a way, it is as if the conditional symbol, “ $|$ ,” defines the *Universe* with which to make the comparisons. (Left) Likewise, the same darker area that was originally part of $B$ represents $P (B | A)$ which makes up 1/4 of the area of $A$ . Thus $P open parenthesis B vertical bar A close parenthesis equals 1 / 4$ .

Venn-style diagram of mutually exclusive statements: rectangles A (1/4 of the Universe) and B (1/8) sit side by side with no overlap at all. — **Figure 1.5.** Venn diagram of mutually exclusive statements. One can see that $P open parenthesis A and B close parenthesis equals 0$ (the overlap is zero) and $P (A or B) = P (A) + P (B)$ (the total area is just the sum of the two areas)

$\begin{matrix} P (A or B) = P (A) + P (B) \end{matrix}$

Further, it is straightforward to see from this diagram the following properties for mutually exclusive events

$\begin{matrix} P (A and B) = 0 \\ P (A | B) = 0 \\ P (B | A) = 0 \end{matrix}$

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.