Lesson 2: Language of Probability

Nicky Wakim

2025-10-01

Learning Objectives

  1. Use set notation, Venn diagrams, and the concepts of unions, intersections, complements, and mutually exclusive events to represent and describe events.

  2. Apply the axioms of probability and related properties to calculate probabilities and prove simple results.

  3. Explain and use De Morgan’s Laws to simplify and solve probability problems.

  4. Connect partitions and all rules of probability to calculate probabilities.

Where are we?

Learning Objectives

  1. Use set notation, Venn diagrams, and the concepts of unions, intersections, complements, and mutually exclusive events to represent and describe events.

  1. Apply the axioms of probability and related properties to calculate probabilities and prove simple results.

  2. Explain and use De Morgan’s Laws to simplify and solve probability problems.

  3. Connect partitions and all rules of probability to calculate probabilities.

Set Theory (1/2)

 

Definition: Union

The union of events \(A\) and \(B\), denoted by \(A \cup B\), contains all outcomes that are in \(A\) or \(B\) or both

Definition: Intersection

The intersection of events \(A\) and \(B\), denoted by \(A \cap B\), contains all outcomes that are both in \(A\) and \(B\).

Venn diagrams

Set Theory (2/2)

 

Definition: Complement

The complement of event \(A\), denoted by \(A^C\) or \(A'\), contains all outcomes in the sample space \(S\) that are not in \(A\) .

Definition: Mutually Exclusive

Events \(A\) and \(B\) are mutually exclusive, or disjoint, if they have no outcomes in common. In this case \(A \cap B = \emptyset\), where \(\emptyset\) is the empty set.

Venn diagrams

Learning Objective

  1. Use set notation, Venn diagrams, and the concepts of unions, intersections, complements, and mutually exclusive events to represent and describe events.
  1. Apply the axioms of probability and related properties to calculate probabilities and prove simple results.

  1. Explain and use De Morgan’s Laws to simplify and solve probability problems.

  2. Connect partitions and all rules of probability to calculate probabilities.

Probability Axioms

Axiom 1

For every event \(A\), \(0\leq\mathbb{P}(A)\leq 1\). Probability is between 0 and 1.

Axiom 2

For the sample space \(S\), \(\mathbb{P}(S)=1\).

Axiom 3

If \(A_1, A_2, A_3, \ldots\), is a collection of disjoint events, then \[\mathbb{P}\Big( \bigcup \limits_{i=1}^{\infty}A_i\Big) = \sum_{i=1}^{\infty}\mathbb{P}(A_i).\] The probability of at least one \(A_i\) is the sum of the individual probabilities of each.

Some probability properties

Using the Axioms, we can prove all other probability properties! Events A, B, and C are not necessarily disjoint!

Proposition 1

For any event \(A\), \(\mathbb{P}(A)= 1 - \mathbb{P}(A^C)\)

Proposition 2

\(\mathbb{P}(\emptyset)=0\)

Proposition 3

If \(A \subseteq B\), then \(\mathbb{P}(A) \leq \mathbb{P}(B)\)

Proposition 4

\[\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)\] where \(A\) and \(B\) are not necessarily disjoint

Proposition 5

\(\begin{aligned} \mathbb{P}(A \cup B & \cup C) = \mathbb{P}(A) + \mathbb{P}(B) + \\ & \mathbb{P}(C) - \mathbb{P}(A \cap B) - \mathbb{P}(A \cap C) - \\ & \mathbb{P}(B \cap C) + \mathbb{P}(A \cap B \cap C) \end{aligned}\)

Proposition 1 Proof

Proposition 1

For any event \(A\), \(\mathbb{P}(A)= 1 - \mathbb{P}(A^C)\)

Use Axioms!

A1: \(0\leq\mathbb{P}(A)\leq 1\)

A2: \(\mathbb{P}(S)=1\)

A3: For disjoint \(A_i\),

\(\mathbb{P}\Big( \bigcup \limits_{i=1}^{\infty}A_i\Big) = \sum_{i=1}^{\infty}\mathbb{P}(A_i)\)

Proposition 2 Proof

Proposition 2

\(\mathbb{P}(\emptyset)=0\)

Use Axioms!

A1: \(0\leq\mathbb{P}(A)\leq 1\)

A2: \(\mathbb{P}(S)=1\)

A3: For disjoint \(A_i\),

\(\mathbb{P}\Big( \bigcup \limits_{i=1}^{\infty}A_i\Big) = \sum_{i=1}^{\infty}\mathbb{P}(A_i)\)

Proposition 3 Proof

Proposition 3

If \(A \subseteq B\), then \(\mathbb{P}(A) \leq \mathbb{P}(B)\)

Use Axioms!

A1: \(0\leq\mathbb{P}(A)\leq 1\)

A2: \(\mathbb{P}(S)=1\)

A3: For disjoint \(A_i\),

\(\mathbb{P}\Big( \bigcup \limits_{i=1}^{\infty}A_i\Big) = \sum_{i=1}^{\infty}\mathbb{P}(A_i)\)

Proposition 4 Visual Proof

Proposition 4

\(\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)\)

Proposition 5 Visual Proof

Proposition 5

\(\mathbb{P}(A \cup B \cup C) = \mathbb{P}(A) + \mathbb{P}(B) + \mathbb{P}(C) - \mathbb{P}(A \cap B) - \mathbb{P}(A \cap C) - \mathbb{P}(B \cap C) + \mathbb{P}(A \cap B \cap C)\)

Some final remarks on these proposition

  • Notice how we spliced events into multiple disjoint events
    • It is often easier to work with disjoint events

 

  • If we want to calculate the probability for one event, we may need to get creative with how we manipulate other events and the sample space
    • Helps us use any incomplete information we have

Learning Objectives

  1. Use set notation, Venn diagrams, and the concepts of unions, intersections, complements, and mutually exclusive events to represent and describe events.

  2. Apply the axioms of probability and related properties to calculate probabilities and prove simple results.

  1. Explain and use De Morgan’s Laws to simplify and solve probability problems.
  1. Connect partitions and all rules of probability to calculate probabilities.

De Morgan’s Laws

Theorem: De Morgan’s 1st Law

For a collection of events (sets) \(A_1, A_2, A_3, \ldots\)

\[\bigcap\limits_{i=1}^{n}A_i^C = \Big(\bigcup\limits_{i=1}^{n}A_i\Big)^C\]

“all not A = \((\)at least one event A\()^C\)” or “intersection of the complements is the complement of the union”

Theorem: De Morgan’s 2nd Law

For a collection of events (sets) \(A_1, A_2, A_3, \ldots\)

\[\bigcup\limits_{i=1}^{n}A_i^C = \Big(\bigcap\limits_{i=1}^{n}A_i\Big)^C\]

“at least one event not A = \((\)all A\()^C\)” or “union of complements is complement of the intersection”

BP example variation (1/3)

  • Suppose you have \(n\) subjects in a study.

  • Let \(H_i\) be the event that person \(i\) has high BP, for \(i=1\ldots n\).

 

Use set theory notation to denote the following events:

  1. Event subject \(i\) does not have high BP

  2. Event all \(n\) subjects have high BP

  3. Event at least one subject has high BP

  4. Event all of them do not have high BP

  5. Event at least one subject does not have high BP

BP example variation (2/3)

  • Suppose you have \(n\) subjects in a study.

  • Let \(H_i\) be the event that person \(i\) has high BP, for \(i=1\ldots n\).

Use set theory notation to denote the following events:

  1. Event subject \(i\) does not have high BP

     

  2. Event all \(n\) subjects have high BP

     

     

  3. Event at least one subject has high BP

BP example variation (3/3)

  1. Event all of them do not have high BP

     

     

     

  2. Event at least one subject does not have high BP

Remarks on De Morgan’s Laws

  • These laws also hold for infinite collections of events.

     

  • Draw Venn diagrams to convince yourself that these are true!

     

  • These laws are very useful when calculating probabilities.

    • This is because calculating the probability of the intersection of events is often much easier than the union of events.

    • This is not obvious right now, but we will see in the coming chapters why.

Learning Objectives

  1. Use set notation, Venn diagrams, and the concepts of unions, intersections, complements, and mutually exclusive events to represent and describe events.

  2. Apply the axioms of probability and related properties to calculate probabilities and prove simple results.

  3. Explain and use De Morgan’s Laws to simplify and solve probability problems.

  1. Connect partitions and all rules of probability to calculate probabilities.

Partitions

Definition: Partition

A set of events \(\{A_i\}_{i=1}^{n}\) create a partition of \(A\), if

  • the \(A_i\)’s are disjoint (mutually exclusive) and

  • \(\bigcup \limits_{i=1}^n A_i = A\)

Example 2

  • If \(A \subset B\), then \(\{A, B \cap A^C\}\) is a partition of \(B\).

  • If \(S = \bigcup \limits_{i=1}^n A_i\), and the \(A_i\)’s are disjoint, then the \(A_i\)’s are a partition of the sample space.

Creating partitions is sometimes used to help calculate probabilities, since by Axiom 3 we can add the probabilities of disjoint events.

Weekly medications

Example 3

If a subject has an

  • 80% chance of taking their medication this week,

  • 70% chance of taking their medication next week, and

  • 10% chance of not taking their medication either week,

then find the probability of them taking their medication exactly one of the two weeks.

Hint: Draw a Venn diagram labelling each of the parts to find the probability.