1.3 Set theory and Venn diagrams

Set Theory and Venn Diagrams

Set theory and Venn diagrams give you the tools to describe how events relate to each other and to calculate probabilities systematically. Almost every probability formula you'll encounter builds on set operations like unions, intersections, and complements, so getting comfortable with these now pays off throughout the course.

Set Theory for Probability

Fundamental Set Operations

A set is just a collection of objects (in probability, these objects are outcomes). Set operations let you combine, overlap, or exclude sets in precise ways.

• Union of sets A and B ($A \cup B$): all elements that are in A, in B, or in both. Think "A or B."
• Intersection of sets A and B ($A \cap B$): only the elements that are in both A and B. Think "A and B."
• Complement of set A ($A^c$): everything in the universal set that is not in A. Think "not A."

These three operations connect directly to probability through the addition rule:

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

You subtract $P(A \cap B)$ because the outcomes in the overlap get counted once in $P(A)$ and again in $P(B)$. Without subtracting, you'd double-count them.

De Morgan's Laws show how complements interact with unions and intersections:

• $(A \cup B)^c = A^c \cap B^c$
• $(A \cap B)^c = A^c \cup B^c$

In words: "not (A or B)" is the same as "not A and not B," and "not (A and B)" is the same as "not A or not B." These come up often when you need to rewrite a probability expression in a more convenient form.

Applications to Probability

The probability of an event A in a finite sample space with equally likely outcomes is:

$P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

When events A and B are mutually exclusive (they can't happen at the same time), their intersection is empty, so the addition rule simplifies to:

$P(A \cup B) = P(A) + P(B)$

For three events, the inclusion-exclusion principle extends the addition rule:

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

You add back the triple intersection because it got subtracted too many times in the pairwise terms.

Conditional probability uses intersection directly:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

This tells you the probability of A given that B has already occurred. You're essentially shrinking the sample space down to just B, then asking how much of B is also in A.

The law of total probability lets you break a hard probability into easier pieces. If $B_1, B_2, \ldots, B_n$ form a partition of the sample space (mutually exclusive and collectively exhaustive):

$P(A) = \sum_{i=1}^{n} P(A \mid B_i) \, P(B_i)$

Bayes' theorem flips a conditional probability around:

$P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B)}$

This is useful when you know $P(B \mid A)$ but need $P(A \mid B)$. A classic example: you know how often a medical test is positive given the disease, and you want to find the probability of having the disease given a positive test.

Relationships Between Sets

Mutually Exclusive and Exhaustive Sets

• Mutually exclusive sets share no elements: $A \cap B = \emptyset$. If one event happens, the other cannot.
• Collectively exhaustive sets cover every outcome in the sample space: their union equals the universal set.
• A partition is a collection of sets that are both mutually exclusive and collectively exhaustive. For example, {even outcomes, odd outcomes} partitions the sample space of a single die roll. Partitions are what make the law of total probability work.

Independence and Exclusivity

Independence and mutual exclusivity are easy to confuse, but they describe very different relationships.

• Independent events satisfy $P(A \cap B) = P(A) \cdot P(B)$. Knowing one occurred doesn't change the probability of the other.
• Mutually exclusive events satisfy $P(A \cap B) = 0$. They cannot both occur.

Notice the tension: if two events with nonzero probability are mutually exclusive, they can't be independent. Knowing one happened tells you the other definitely didn't, which changes its probability to zero.

Visualizing Events with Venn Diagrams

Constructing Venn Diagrams

Venn diagrams use overlapping circles inside a rectangle to show how sets relate.

• The rectangle represents the universal set (the entire sample space).
• Each circle represents an event.
• The overlap between circles represents the intersection.
• The area inside the rectangle but outside a circle represents that event's complement.

For two sets, a Venn diagram creates four distinct regions: only A, only B, both A and B, and neither. For three sets, you get eight distinct regions (including the triple overlap in the center and the region outside all three circles).

Shading or labeling each region with its probability makes it much easier to keep track of what you're calculating.

Interpreting Venn Diagrams

• Circles that don't overlap at all represent mutually exclusive events.
• A circle entirely inside another circle means one event is a subset of the other (whenever the inner event occurs, the outer one does too).
• The area of each region, relative to the total, can represent the probability of that combination of events.

Venn diagrams are especially helpful when a problem gives you several overlapping categories. For instance, if a survey asks customers whether they shop online, shop in-store, or belong to a loyalty program, a three-set Venn diagram lets you place each customer in exactly one of the eight regions and read off any probability you need.

Probability Calculations with Sets

Basic Probability Calculations

Here are the core formulas, collected in one place:

• Complement rule: $P(A^c) = 1 - P(A)$
• Addition rule (general): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Addition rule (mutually exclusive): $P(A \cup B) = P(A) + P(B)$

The complement rule is surprisingly useful. Whenever "at least one" appears in a problem, it's often easier to calculate the probability that none occur and subtract from 1.

Advanced Probability Techniques

Conditional probability narrows your focus to a specific region of the Venn diagram:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

On a Venn diagram, this means you ignore everything outside circle B and ask what fraction of B's area is also inside A.

Law of total probability for a partition $B_1, B_2, \ldots, B_n$:

$P(A) = P(A \mid B_1)P(B_1) + P(A \mid B_2)P(B_2) + \cdots + P(A \mid B_n)P(B_n)$

Bayes' theorem for updating probabilities with new evidence:

$P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B)}$

A quick example to see Bayes' theorem in action: suppose a disease affects 1% of a population, and a test is 95% accurate (both for true positives and true negatives). If someone tests positive, Bayes' theorem shows the probability they actually have the disease is only about 16%, not 95%. The low base rate of the disease matters enormously. Working through examples like this with a Venn diagram or a probability tree makes the formula much more intuitive.